Hyperoperations Wiki - User contributions [en]

File:Fps9 stb.png

2022-07-21T20:07:36Z

Bo198214: Shows the fixed points of b^z for b moving along the Shell-Thron boundary

Shows the fixed points of b^z for b moving along the Shell-Thron boundary

Shell-Thron region

2019-07-16T19:09:38Z

Bo198214:

The Shell-Thron region is the region of complex numbers $b$ where the limit of the infinite exponential tower
$$\lambda=\lim_{n\to\infty} \exp_b^{\circ n}(1)$$
exists; to be explicit we mean the main branch of the power: $\exp_b(z):=b^z:=\exp(\log(b)z)$.

If this limit $\lambda$ exists then must $b^\lambda=\lambda$, i.e. $\lambda$ must be a fixpoint of $f(z)=b^z$, or
$$\lambda=\frac{W(-\log(b))}{-\log(b)}$$

By a result of Barrow the limit $\lambda$ exists exactly for all $b$ where $|\log(\lambda)|\le 1$.

The boundary of the Thron-Shell region, i.e. all bases $b$ where $|\log(\lambda)|=1$, is given by all $b$ where $|\exp_b'(\lambda)|=1$ (i.e. lambda is a parabolic fixpoint of $\exp_b$) due to the following rearrangements:
$$\begin{align*}
1=|\exp_b'(\lambda)|=|\log(b)\exp_b(\lambda)|=|\log(b)\lambda|=|\log(b^\lambda)|=|\log(\lambda)|
\end{align*}$$

These rearrangements are not proper derivations, but hints only, as with complex bases one always has to worry about landing in the right branch.

Following from this the boundary can be given as a curve $\phi\mapsto \exp(e^{i\phi-e^{i\phi}}),\phi\in[0,2\pi)$.

<hr>
The following picture of the upper part of the Shell-Thron region was made by [[User:Andydude|Andrew]].

[[File:Shell-region.png]]

<hr>

This picture gives a combined view of the locus of the fixpoints, their logs (being on the unit-circle) and the corresponding complex bases: (the symbol $\lambda$ is replaced by the letter $t$ )
[[File:ShellThron_utb.png|600px]]

== Relevant posts on the Tetrationforum ==

* [http://math.eretrandre.org/tetrationforum/showthread.php?tid=263&pid=3011#pid3011 Question about infinite tetrate]

* [http://math.eretrandre.org/tetrationforum/showthread.php?tid=120&pid=1377#pid1377 Infinite tetration of the imaginary unit]

Shell-Thron region

2019-07-16T19:08:44Z

Bo198214:

The Shell-Thron region is the region of complex numbers $b$ where the limit of the infinite exponential tower
$$\lambda=\lim_{n\to\infty} \exp_b^{\circ n}(1)$$
exists; to be explicit we mean the main branch of the power: $\exp_b(z):=b^z:=\exp(\log(b)z)$.

If this limit $\lambda$ exists then must $b^\lambda=\lambda$, i.e. $\lambda$ must be a fixpoint of $f(z)=b^z$, or
$$\lambda=\frac{W(-\log(b))}{-\log(b)}$$

By a result of Barrow the limit $\lambda$ exists exactly for all $b$ where $|\log(\lambda)|\le 1$.

The boundary of the Thron-Shell region, i.e. all bases $b$ where $|\log(\lambda)|=1$, is given by all $b$ where $|\exp_b'(\lambda)|=1$ (i.e. lambda is a parabolic fixpoint of $\exp_b$) due to the following rearrangements:
$$\begin{align*}
1=|\exp_b'(\lambda)|=|\log(b)\exp_b(\lambda)|=|\log(b)\lambda|=|\log(b^\lambda)|=|\log(\lambda)|
\end{align*}$$

These rearrangements are not proper derivations, but hints only, as with complex bases one always has to worry about landing in the right branch.

Following from this the boundary can be given as a curve $\phi\mapsto \exp(e^{i\phi-e^{i\phi}}),\phi\in(0,2\pi)$.

<hr>
The following picture of the upper part of the Shell-Thron region was made by [[User:Andydude|Andrew]].

[[File:Shell-region.png]]

<hr>

This picture gives a combined view of the locus of the fixpoints, their logs (being on the unit-circle) and the corresponding complex bases: (the symbol $\lambda$ is replaced by the letter $t$ )
[[File:ShellThron_utb.png|600px]]

== Relevant posts on the Tetrationforum ==

* [http://math.eretrandre.org/tetrationforum/showthread.php?tid=263&pid=3011#pid3011 Question about infinite tetrate]

* [http://math.eretrandre.org/tetrationforum/showthread.php?tid=120&pid=1377#pid1377 Infinite tetration of the imaginary unit]

Shell-Thron region

2019-07-16T19:08:08Z

Bo198214:

The Shell-Thron region is the region of complex numbers $b$ where the limit of the infinite exponential tower
$$\lambda=\lim_{n\to\infty} \exp_b^{\circ n}(1)$$
exists; to be explicit we mean the main branch of the power: $\exp_b(z):=b^z:=\exp(\log(b)z)$.

If this limit $\lambda$ exists then must $b^\lambda=\lambda$, i.e. $\lambda$ must be a fixpoint of $f(z)=b^z$, or
$$\lambda=\frac{W(-\log(b))}{-\log(b)}$$

By a result of Barrow the limit $\lambda$ exists exactly for all $b$ where $|\log(\lambda)|\le 1$.

The boundary of the Thron-Shell region, i.e. all bases $b$ where $|\log(\lambda)|=1$, is given by all $b$ where $|\exp_b'(\lambda)|=1$ (i.e. lambda is a parabolic fixpoint of $\exp_b$) due to the following rearrangements:
$$\begin{align*}
1=|\exp_b'(\lambda)|=|\log(b)\exp_b(\lambda)|=|\log(b)\lambda|=|\log(b^\lambda)|=|\log(\lambda)|
\end{align*}$$

These rearrangements are not proper derivations, but hints only, as with complex bases one always has to worry about landing in the right branch.

Following from this the boundary can be given as a curve $\phi\mapsto exp(e^{i\phi-e^{i\phi}}),\phi\in(0,2\pi)$.

<hr>
The following picture of the upper part of the Shell-Thron region was made by [[User:Andydude|Andrew]].

[[File:Shell-region.png]]

<hr>

This picture gives a combined view of the locus of the fixpoints, their logs (being on the unit-circle) and the corresponding complex bases: (the symbol $\lambda$ is replaced by the letter $t$ )
[[File:ShellThron_utb.png|600px]]

== Relevant posts on the Tetrationforum ==

* [http://math.eretrandre.org/tetrationforum/showthread.php?tid=263&pid=3011#pid3011 Question about infinite tetrate]

* [http://math.eretrandre.org/tetrationforum/showthread.php?tid=120&pid=1377#pid1377 Infinite tetration of the imaginary unit]

Shell-Thron region

2019-07-16T18:41:59Z

Bo198214:

The Shell-Thron region is the region of complex numbers $b$ where the limit of the infinite exponential tower
$$\lambda=\lim_{n\to\infty} \exp_b^{\circ n}(1)$$
exists; to be explicit we mean the main branch of the power: $\exp_b(z):=b^z:=\exp(\log(b)z)$.

If this limit $\lambda$ exists then must $b^\lambda=\lambda$, i.e. $\lambda$ must be a fixpoint of $f(z)=b^z$, or
$$\lambda=\frac{W(-\log(b))}{-\log(b)}$$

By a result of Barrow the limit $\lambda$ exists exactly for all $b$ where $|\log(\lambda)|\le 1$.

The boundary of the Thron-Shell region, i.e. all bases $b$ where $|\log(\lambda)|=1$, is given by all $b$ where $|\exp_b'(\lambda)|=1$ (i.e. lambda is a parabolic fixpoint of $\exp_b$) due to the following rearrangements:
$$\begin{align*}
1=|\exp_b'(\lambda)|=|\log(b)\exp_b(\lambda)|=|\log(b)\lambda|=|\log(b^\lambda)|=|\log(\lambda)|
\end{align*}$$

These rearrangements are not proper derivations, but hints only, as with complex bases one always has to worry about landing in the right branch.

The boundary can also be given by the formula $|-W(-\log(b))|=1$.

<hr>
The following picture of the upper part of the Shell-Thron region was made by [[User:Andydude|Andrew]].

[[File:Shell-region.png]]

<hr>

This picture gives a combined view of the locus of the fixpoints, their logs (being on the unit-circle) and the corresponding complex bases: (the symbol $\lambda$ is replaced by the letter $t$ )
[[File:ShellThron_utb.png|600px]]

== Relevant posts on the Tetrationforum ==

* [http://math.eretrandre.org/tetrationforum/showthread.php?tid=263&pid=3011#pid3011 Question about infinite tetrate]

* [http://math.eretrandre.org/tetrationforum/showthread.php?tid=120&pid=1377#pid1377 Infinite tetration of the imaginary unit]

Shishikura perturbed Fatou coordinates

2019-07-16T08:42:42Z

Bo198214:

Shishikura writes<ref name="Shishikura2000">Shishikura, M. (2000). Bifurcation of parabolic fixed points. In Lei, Tan, The Mandelbrot set, theme and variations. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 274, 325-363 (2000)</ref> the following

== $\mathcal{F}_0$ ==
[p. 327]

If $f_0''(0)\neq 0$ by another coordinate change we may assume that $f_0''(0)=1$, so define
$$ \mathcal{F}_0 = \{ f_0 \colon f_0 \quad\text{is defined and analytic in a neighborhood of 0}\quad f_0(0)=0, f_0'(0)=1, f_0''(0)=1\}$$

== Neighborhood of $f$ in the compact-open topology with domain of definition ==
[ p. 332 ]

In what follows a map $f$ is always associated with its domain of definition $D(f)$. That is two maps are considered as distinct if they have distinct domains, even if the one is an extension of the other. A neighborhood of an analytic map $f\colon D(f)\to\overline{\C}$ is a set containing
$$\{ g \colon g \quad\text{is analytic},\quad D(g)\supset K, \sup_{z\in K} d(f(z)-g(z))<\eps \} $$
where $K$ is a compact set in $D(f)$, $\eps>0$ and $d(.,.)$ is the spherical metric.
A sequence $(f_n)$ converges to $f$ in this topology if and only if for any compact set $K\subset D(f)$ there exists $n_0$ such that $K\subset D(f_n)$ for all $n\ge n_0$ and $f_n|_K\to f|_K$ uniformly on $K$ as $n\to\infty$. The system of these neighborhoods define the "compact-open topology together with the domain of defintion", which is unfortunately not Hausdorff, since an extension of $f$ is contained in any neighborhood of $f$.

For $b_1,b_2\in\C$ with $\Re(b_1)<\Re(b_2)$ define
$$Q(b_1,b_2)=\{z\in\C\colon \Re(z-b_1)>-|\Im(z-b_1)|, \Re(z-b_2)>|\Im(z-b_2)|\}$$
If $b_1=-\infty$ (resp. $b_2=\infty$) then the corresponding condition should be removed.

== Proposition 2.5.2 ==
[p. 333]

Let $f$ be a holomorphic function defined on $Q=Q(b_1,b_2)$ with $\Re(b_2)>\Re(b_1)+2$ (here $b_1$ or $b_2$ may be $-\infty$ or $\infty$). Suppose
$$|F(z)-(z+1)|<\frac{1}{4},\quad |F'(z)-(z+1)|<\frac{1}{4}\quad \text{for all}\quad z\in Q.$$

Then
# $F$ is univalent [means injective] on $Q$.
# Let $z_0\in Q$ be a point such that $\Re(b_1)<\Re(z_0)<\Re(b_2)-\frac{5}{4}$. Denote by $S$ the closed region (a strip) bounded by the two curves $\ell=\{z_0+i y\colon y\in\R\}$ and $F(\ell)$. Then for any $z\in Q$ there exist a unique $n\in \Z$ such that $F^n(z)$ is defined and belongs to $S-F(\ell)$.
# There exists an univalent function $\Phi\colon Q\to \C$ satisfying $$\Phi(F(z))=\Phi(z)+1$$ whenever both sides are defined. Moreover $\Phi$ is unique up to an addition of a constant.
# Fix a point $z_0\in Q$. If we normalize $\Phi$ by $\Phi(z_0)=0$ then the correspondence $F\mapsto \Phi$ is continuous with respect to the compact-open topology.

== $\mathcal{F}$, $\mathcal{F}_1$ and $\alpha(f)$ ==
[p. 339]

$$ \mathcal{F} = \{ f \colon f \quad\text{is defined and analytic in a neighborhood of 0}\quad f(0)=0, f_0'(0)\neq 0\}$$
For $f\in \mathcal{F}$ we express the derivative
$$f'(0)=\exp(2\pi i \alpha(f))$$
where $\alpha(f)\in\C$ and $-\frac{1}{2} < \Re(\alpha(f)) \le \frac{1}{2}$ our study is focussed on the maps in the following class
$$ \mathcal{F}_1 = \{ f\in \mathcal{F}\colon |\arg(\alpha(f))| < \frac{\pi}{4}\} $$

=== Comments ===

By the definition we have $$
\begin{align*}
\log(f'(0))&=2\pi i\alpha(f)=2\pi i re^{i\beta}=2\pi r e^{i(\beta+\frac{\pi}{2})}\\
\arg(\log(f'(0))&=\beta+\frac{\pi}{2}
\end{align*}$$
so the condition for $\mathcal{F}_1$ is
$|\arg(\log(f'(0)))-\frac{\pi}{2}|<\frac{\pi}{4}$ or
$$\frac{\pi}{4} < \arg(\log(f'(0)) < \frac{3\pi}{4}$$

== $\sigma(f)$ ==
[p. 340]

Let $\sigma(f)$ be the other fixpoint of $f$ (besides 0).

...

$$\sigma(f) = 2\pi i \alpha(f)(1+o(1)), \quad\text{as}\quad f\to f_0$$

=== Comments ===

By the comment before, we can say that $\sigma(f)$ lies roughly in the sector of angle $(\frac{\pi}{4},\frac{3\pi}{4})$ above the tip 0.

== Proposition 3.2.2 ==
[p. 340]

Let $f_0\in\mathcal{F}_0$. There exists a neighborhood $\mathcal{N}_0$ of $f_0$ (in the sense of the topology defined in 2.5.1) such that if $f\in\mathcal{N}_0\cap \mathcal{F}_1$, then there exist closed Jordan domains $S_{-,f}$, $S_{+,f}$ and analytic functions $\Phi_{-,f}$, $\Phi_{+,f}$ satisfying the following:

# $S_{\pm,f}$ is bounded by an arc $\ell_{\pm}$ and its image $f(\ell_{\pm})$ such that $\ell_{\pm}$ joins two fixed points $\{0,\sigma(f)\}$, $\ell_{\pm}\cap f(\ell_{\pm}) = \{0,\sigma(f)\}$ and $S_{-,f}\cap S_{+,f} = \{0,\sigma(f)\}$;
# $\Phi_{\pm,f}$ is defined, analytic and injective in a neighborhood of $S'_{\pm,f} = S_{\pm,f}\setminus \{0,\sigma(f)\}$;
# if $z\in S'_{-,f}$ then there is an $n\ge 1$ such that $f^n(z)\in S'_{+,f}$ and for the smallest such $n$, \[ \Phi_{+,f}(f^n(z)) = \Phi_{-,f}(z) - \frac{1}{\alpha(f)} + n \]
# when $f\in\mathcal{F}_1$ tends to $f_0$, $S_{\pm,f}\to S_{\pm,0}\cup \{0\}$ in the Hausdorff metric, and $\Phi_{\pm,f}\to \Phi_{\pm,0}$ in the topology defined in 2.5.1.

== Proposition 4.4.1 ==
[p. 356]

Let $f_0\in\mathcal{F}_0$, there exists a neighborhood $\mathcal{N}_0$ of $f_0$ (in the previously described "compact-open topology with domain of definition"), such that if $f\in \mathcal{N}_0 \cap \mathcal{F}_1$, then there exist large $\xi_0, \eta_0>0$, and analytic maps $\ph_f\colon D(\ph_f)\to\overline{\C}$ and $\tilde{\mathcal{R}}_f\colon D(\tilde{\mathcal{R}}_f)\to \C$ satisfying the following:
# $D(\ph_f)$ contains the set $$ \mathcal{Q}_f = \{ w\in\C\colon |\arg(-w-\xi_0)| < \frac{2\pi}{3}, |\arg(w+\frac{1}{\alpha(f)}-\xi_0)|<\frac{2\pi}{3} \} $$ and $\ph_f(D(\ph_f)) \subseteq D(f)$; If $w, w+1\in D(\ph_f)$ then $$ \ph_f(w+1)=f(\ph_f(w)) $$ $\ph_f(w)\to 0$ when $w\in \mathcal{Q}_f$ and $\Im(w)\to\infty$; $\ph_f(w)\to\sigma(f)$ when $w\in\mathcal{Q}_f$ and $\Im(w)\to -\infty$.
# $D(\tilde{\mathcal{R}}_f)$ contains $\{ w\in\C\colon |\Im(w)| > \eta_0\}$ and is invariant under $T(w)=w+1$; if $w\in D(\tilde{\mathcal{R}}_f)$ then $$\tilde{\mathcal{R}}_f(w+1)=\tilde{\mathcal{R}}(w)+1;$$ $\tilde{\mathcal{R}}_f(w)-w$ tends to $-\frac{1}{\alpha(f)}$ when $\Im(w)\to\infty$; to a constant when $\Im(w)\to -\infty$.
# If $w\in D(\tilde{\mathcal{R}}_f)\cap D(\ph_f)$ and $w'=\tilde{\mathcal{R}}_f(w)+n\in D(\ph_f)$ for some $n\in\Z$ then $$f^n(\ph_f(w))=\ph_f(w')\quad \text{for}\quad n\ge 0, \quad f^{-n}(\ph_f(w'))=\ph_f(w)\quad \text{for}\quad n<0$$ Moreover if $|\arg(w'+\frac{1}{2\alpha(f)}-\xi_0)|<\frac{2\pi}{3}$ then $n>0$.
# When $f\in\mathcal{F_1}$ and $f\to f_0$ $$\ph_f\to\ph_0\quad\text{and}\quad \tilde{\mathcal{R}}_f + \frac{1}{\alpha(f)} \to \tilde{\mathcal{E}}_{f_0}$$

== Proposition A.1 ==
[p. 360]

In the settings of Proposition 3.2.2 (resp. 2.5.2), assume in addtion that $f_s(z)$ (resp. $F_s(z)$) is a family of holomorphic maps for $s\in \Delta \cup \{0\} \subset \mathbb{C}^k$, with $\Delta\subset\mathbb{C}^k$ an open connected set containing 0 in the boundary, such that $f_s$ (resp. $F_s(z)$) is continuous for $s\in\Delta\cup \{0\}$ and analytic in $s\in\Delta$, then the Fatou coordinates $\Phi_{\pm,f_s}$ (resp. $\Phi_{F_s}$) depend analytically on $s$ for $s\in\Delta$.

<references/>

Uniqueness of Tetration

2017-01-04T08:04:23Z

Bo198214: work in progress

'''This is work in progress, dont rely on it!'''

== Proposition ==
For each $b>e^{1/e}$ there exists exactly one holomorphic function $\sigma_b$ on $D=\C\setminus\{x \le -2\}$ which is real for all $x>-2$ and satisfies for all $z\in D$:
$$\sigma_b(0)=1, \sigma_b(z+1)=b^{\sigma_b(z)}$$ and $$\sup_{t\to\infty}\left| \sigma_b({\rm i}t)\right| <\infty$$.

==== Proof. ====
We know there exists already a solution $\tau_b$ which satisfies all conditions and its inverse $\alpha_b$ which is a whole function and satisifies $\alpha_b(b^z)=\alpha_b(z)+1$.
Then we know that
$$\sup_{t\to\infty}\alpha_b(\sigma_b({\rm i}t))<\infty$$
and the function $\delta(z)=\alpha_b(\sigma_b(z))-z$, holomorphic on $D$, is periodic with period 1:
$$\delta(z+1)=(\alpha_b(\sigma_b(z+1))-(z+1)=\alpha_b(\sigma_b(z))-z=\delta_b(z)$$
and is real for $z > -2$ (and can be continued to $\R$) and can so be developed into a real Fourier-Series ($A_k$, $\phi_k$ in $\R$):

$$\delta(t)=\sum_{k=0}^{\infty}A_{k}\cos\left(2\pi kt-\varphi_{k}\right)$$

$$\delta(z)=\sum_{k=0}^{\infty}\frac{A_{k}}{2}\left(\exp\left(2\pi {\rm i}kz-{\rm i}\varphi_{k}\right)+\exp\left(-2\pi {\rm i}kz+{\rm i}\varphi_{k}\right)\right)$$
$$\delta(x+{\rm i}y)=\sum_{k=0}^{\infty}\frac{A_{k}}{2}\left(\exp\left({\rm i}(2\pi kx-\varphi_{k})-2\pi k y\right)+\exp\left({\rm i}(-2\pi kx+\varphi_{k})+2\pi k y\right)\right)
=\sum_{k=0}^{\infty}\frac{A_{k}}{2}\left(\cos(2\pi kx-\varphi_{k})\left(\exp\left(-2\pi k y\right)+\exp(2\pi ky)\right)+{\rm i}\sin(2\pi kx-\varphi_{k})\left(-\exp\left(2\pi k y\right)+\exp(-2\pi ky)\right)\right)$$

$$\Im(\delta(x+{\rm i}y)+x+{\rm i}y)=y+\sum_{k=0}^{\infty}\frac{A_{k}}{2}\left(-\exp\left(2\pi k y\right)+\exp(-2\pi ky)\right)\sin(2\pi kx-\varphi_{k})$$

$$
\alpha_b(\sigma_b({\rm i}t))=\delta({\rm i}t)+{\rm i}t={\rm i}t+\sum_{k=0}^{\infty}\frac{A_{k}}{2}\left(\exp\left(-2\pi kt -{\rm i}\varphi_{k}\right)+\exp\left(2\pi kt+{\rm i}\varphi_{k}\right)\right)
$$

But this expression can only be bounded with respect to $t$ if $A_k=0$ for all $k\ge 1$. Hence $\delta$ is a constant $c$ and $\sigma_b(z)=\tau_b(z+c)$.

Uniqueness of Tetration

2017-01-03T19:44:58Z

Bo198214: /* Proposition */

This is work in progress, dont rely on it!

== Proposition ==
For each $b>e^{1/e}$ there exists exactly one holomorphic function $\sigma_b$ on $D=\C\setminus\{x \le -2\}$ which is real for all $x>-2$ and satisfies for all $z\in D$:
$$\sigma_b(0)=1, \sigma_b(z+1)=b^{\sigma_b(z)}$$ and $$\sup_{t\to\infty}\left| \sigma_b({\rm i}t)\right| <\infty$$.

==== Proof. ====
We know there exists already a solution $\tau_b$ which satisfies all conditions and its inverse $\alpha_b$ which is a whole function and satisifies $\alpha_b(b^z)=\alpha_b(z)+1$.
Then we know that
$$\sup_{t\to\infty}\alpha_b(\sigma_b({\rm i}t))<\infty$$
and the function $\delta(z)=\alpha_b(\sigma_b(z))-z$, holomorphic on $D$, is periodic with period 1:
$$\delta(z+1)=(\alpha_b(\sigma_b(z+1))-(z+1)=\alpha_b(\sigma_b(z))-z=\delta_b(z)$$
and is real for $z > -2$ (and can be continued to $\R$) and can so be developed into a real Fourier-Series ($A_k$, $\phi_k$ in $\R$):

$$\delta(t)=\sum_{k=0}^{\infty}A_{k}\cos\left(2\pi kt-\varphi_{k}\right)$$

$$\delta(z)=\sum_{k=0}^{\infty}\frac{A_{k}}{2}\left(\exp\left(2\pi {\rm i}kz-{\rm i}\varphi_{k}\right)+\exp\left(-2\pi {\rm i}kz+{\rm i}\varphi_{k}\right)\right)$$
$$\delta(x+{\rm i}y)=\sum_{k=0}^{\infty}\frac{A_{k}}{2}\left(\exp\left({\rm i}(2\pi kx-\varphi_{k})-2\pi k y\right)+\exp\left({\rm i}(-2\pi kx+\varphi_{k})+2\pi k y\right)\right)
=\sum_{k=0}^{\infty}\frac{A_{k}}{2}\left(\cos(2\pi kx-\varphi_{k})\left(\exp\left(-2\pi k y\right)+\exp(2\pi ky)\right)+{\rm i}\sin(2\pi kx-\varphi_{k})\left(-\exp\left(2\pi k y\right)+\exp(-2\pi ky)\right)\right)$$

$$\Im(\delta(x+{\rm i}y)+x+{\rm i}y)=y+\sum_{k=0}^{\infty}\frac{A_{k}}{2}\left(-\exp\left(2\pi k y\right)+\exp(-2\pi ky)\right)\sin(2\pi kx-\varphi_{k})$$

$$
\alpha_b(\sigma_b({\rm i}t))=\delta({\rm i}t)+{\rm i}t={\rm i}t+\sum_{k=0}^{\infty}\frac{A_{k}}{2}\left(\exp\left(-2\pi kt -{\rm i}\varphi_{k}\right)+\exp\left(2\pi kt+{\rm i}\varphi_{k}\right)\right)
$$

But this expression can only be bounded with respect to $t$ if $A_k=0$ for all $k\ge 1$. Hence $\delta$ is a constant $c$ and $\sigma_b(z)=\tau_b(z+c)$.

Uniqueness of Tetration

2017-01-03T19:44:25Z

Bo198214:

== Proposition ==
For each $b>e^{1/e}$ there exists exactly one holomorphic function $\sigma_b$ on $D=\C\setminus\{x \le -2\}$ which is real for all $x>-2$ and satisfies for all $z\in D$:
$$\sigma_b(0)=1, \sigma_b(z+1)=b^{\sigma_b(z)}$$ and $$\sup_{t\to\infty}\left| \sigma_b({\rm i}t)\right| <\infty$$.

==== Proof. ====
We know there exists already a solution $\tau_b$ which satisfies all conditions and its inverse $\alpha_b$ which is a whole function and satisifies $\alpha_b(b^z)=\alpha_b(z)+1$.
Then we know that
$$\sup_{t\to\infty}\alpha_b(\sigma_b({\rm i}t))<\infty$$
and the function $\delta(z)=\alpha_b(\sigma_b(z))-z$, holomorphic on $D$, is periodic with period 1:
$$\delta(z+1)=(\alpha_b(\sigma_b(z+1))-(z+1)=\alpha_b(\sigma_b(z))-z=\delta_b(z)$$
and is real for $z > -2$ (and can be continued to $\R$) and can so be developed into a real Fourier-Series ($A_k$, $\phi_k$ in $\R$):

$$\delta(t)=\sum_{k=0}^{\infty}A_{k}\cos\left(2\pi kt-\varphi_{k}\right)$$

$$\delta(z)=\sum_{k=0}^{\infty}\frac{A_{k}}{2}\left(\exp\left(2\pi {\rm i}kz-{\rm i}\varphi_{k}\right)+\exp\left(-2\pi {\rm i}kz+{\rm i}\varphi_{k}\right)\right)$$
$$\delta(x+{\rm i}y)=\sum_{k=0}^{\infty}\frac{A_{k}}{2}\left(\exp\left({\rm i}(2\pi kx-\varphi_{k})-2\pi k y\right)+\exp\left({\rm i}(-2\pi kx+\varphi_{k})+2\pi k y\right)\right)
=\sum_{k=0}^{\infty}\frac{A_{k}}{2}\left(\cos(2\pi kx-\varphi_{k})\left(\exp\left(-2\pi k y\right)+\exp(2\pi ky)\right)+{\rm i}\sin(2\pi kx-\varphi_{k})\left(-\exp\left(2\pi k y\right)+\exp(-2\pi ky)\right)\right)$$

$$\Im(\delta(x+{\rm i}y)+x+{\rm i}y)=y+\sum_{k=0}^{\infty}\frac{A_{k}}{2}\left(-\exp\left(2\pi k y\right)+\exp(-2\pi ky)\right)\sin(2\pi kx-\varphi_{k})$$

$$
\alpha_b(\sigma_b({\rm i}t))=\delta({\rm i}t)+{\rm i}t={\rm i}t+\sum_{k=0}^{\infty}\frac{A_{k}}{2}\left(\exp\left(-2\pi kt -{\rm i}\varphi_{k}\right)+\exp\left(2\pi kt+{\rm i}\varphi_{k}\right)\right)
$$

But this expression can only be bounded with respect to $t$ if $A_k=0$ for all $k\ge 1$. Hence $\delta$ is a constant $c$ and $\sigma_b(z)=\tau_b(z+c)$.

Uniqueness of Tetration

2017-01-03T18:55:13Z

Bo198214: /* Proof. */

== Proposition ==
For each $b>e^{1/e}$ there exists exactly one holomorphic function $\sigma_b$ on $D=\C\setminus\{x \le -2\}$ which is real for all $x>-2$ and satisfies for all $z\in D$:
$$\sigma_b(0)=1, \sigma_b(z+1)=b^{\sigma_b(z)}$$ and $$\sup_{t\to\infty}\left| \sigma_b({\rm i}t)\right| <\infty$$.

==== Proof. ====
We know there exists already a solution $\tau_b$ which satisfies all conditions and its inverse $\alpha_b$ which is a whole function and satisifies $\alpha_b(b^z)=\alpha_b(z)+1$.
Then we know that
$$\sup_{t\to\infty}\alpha_b(\sigma_b({\rm i}t))<\infty$$
and the function $\delta(z)=\alpha_b(\sigma_b(z))-z$, holomorphic on $D$, is periodic with period 1:
$$\delta(z+1)=(\alpha_b(\sigma_b(z+1))-(z+1)=\alpha_b(\sigma_b(z))-z=\delta_b(z)$$
and is real for $z > -2$ (and can be continued to $\R$) and can so be developed into a real Fourier-Series ($A_k$, $\phi_k$ in $\R$):

$$\delta(t)=\sum_{k=0}^{\infty}A_{k}\cos\left(2\pi kt-\varphi_{k}\right)$$

$$\delta(z)=\sum_{k=0}^{\infty}\frac{A_{k}}{2}\left(\exp\left(2\pi {\rm i}kz-{\rm i}\varphi_{k}\right)+\exp\left(-2\pi {\rm i}kz+{\rm i}\varphi_{k}\right)\right)$$
$$\delta(x+{\rm i}y)=\sum_{k=0}^{\infty}\frac{A_{k}}{2}\left(\exp\left({\rm i}(2\pi kx-\varphi_{k})-2\pi k y\right)+\exp\left({\rm i}(-2\pi kx+\varphi_{k})+2\pi k y\right)\right)
=\sum_{k=0}^{\infty}\frac{A_{k}}{2}\left(\cos(2\pi kx-\varphi_{k})\left(\exp\left(-2\pi k y\right)+\exp(2\pi ky)\right)+{\rm i}\sin(2\pi kx-\varphi_{k})\left(-\exp\left(2\pi k y\right)+\exp(-2\pi ky)\right)\right)$$

$$\Im(\delta(x+{\rm i}y))=\sum_{k=0}^{\infty}\frac{A_{k}}{2}\sin(2\pi kx-\varphi_{k})\left(-\exp\left(2\pi k y\right)+\exp(-2\pi ky)\right)$$

$$
\alpha_b(\sigma_b({\rm i}t))=\delta({\rm i}t)+{\rm i}t={\rm i}t+\sum_{k=0}^{\infty}\frac{A_{k}}{2}\left(\exp\left(-2\pi kt -{\rm i}\varphi_{k}\right)+\exp\left(2\pi kt+{\rm i}\varphi_{k}\right)\right)
$$

But this expression can only be bounded with respect to $t$ if $A_k=0$ for all $k\ge 1$. Hence $\delta$ is a constant $c$ and $\sigma_b(z)=\tau_b(z+c)$.

Uniqueness of Tetration

2017-01-03T16:42:42Z

Bo198214: /* Proposition */

Uniqueness of Tetration

2017-01-03T16:25:12Z

Bo198214: /* Proof. */

== Proposition ==
For each $b>e^{1/e}$ there exists exactly one holomorphic function $\sigma_b$ on $D=\C\setminus\{x \le -2\}$ which satisfies for all $z\in D$:
$$\overline{\sigma_b(z)}=\sigma_b(\overline{z})$$, $$\sigma_b(z+1)=b^{\sigma_b(z)}$$ and $$\sup_{t\to\infty}\left| \sigma_b(s+{\rm i}t)\right| <\infty$$ for each $s>-2$.

==== Proof. ====
We know there exists already a solution $\tau_b$ which satisfies all conditions and its inverse $\alpha_b$ which is a whole function and satisifies $\alpha_b(b^z)=\alpha_b(z)+1$.
Then we know that the function $\delta(z)=\alpha_b(\sigma_b(z))-z$, holomorphic on $D$, is periodic with period 1:
$$\delta(z+1)=(\alpha_b(\sigma_b(z+1))-(z+1)=\alpha_b(\sigma_b(z))-z=\delta_b(z)$$
and is real for $z > -2$ (and can be continued to $\R$) and can to developed into a real Fourier-Series ($A_k$, $\phi_k$ in $\R$):

$$\delta(t)=\sum_{k=0}^{\infty}A_{k}\cos\left(2\pi kt-\varphi_{k}\right)$$

$$\delta(z)=\sum_{k=0}^{\infty}\frac{A_{k}}{2}\left(\exp\left(2\pi {\rm i}kz-{\rm i}\varphi_{k}\right)+\exp\left(-2\pi {\rm i}kz+{\rm i}\varphi_{k}\right)\right)$$

$$
\delta(s+{\rm i}t)=\sum_{k=0}^{\infty}\frac{A_{k}}{2}\left(\exp\left(2\pi {\rm i}ks-2\pi kt -{\rm i}\varphi_{k}\right)+\exp\left(-2\pi {\rm i}ks+2\pi kt+{\rm i}\varphi_{k}\right)\right)
$$

But this expression can only be bounded with respect to $t$ if $A_k=0$ for all $k\ge 1$. Hence $\delta$ is a constant $c$ and $\sigma_b(z)=\tau_b(z+c)$.

Uniqueness of Tetration

2017-01-03T16:20:22Z

Bo198214: Created page with "== Proposition == For each $b>e^{1/e}$ there exists exactly one holomorphic function $\sigma_b$ on $D=\C\setminus\{x \le -2\}$ which satisfies for all $z\in D$: $$\overline{\s..."

Schroeder iterate

2013-05-01T16:02:59Z

Bo198214: corrected redirect

#REDIRECT[[Schröder iterate]]

Schroeder iterate

2013-05-01T16:02:06Z

Bo198214: Weiterleitung

#REDIRECT[[Schröder Iterate]]

Iteration of fractional linear maps

2013-05-01T15:56:40Z

Bo198214: /* Iterates */

Our concern here are functions of the form $f(z)=\frac{az+b}{cz+d}$ with $c\neq 0$.

== Fixpoints ==
In the case $c\neq 0$, the fixpoints are given by

$L_{1,2}=-\frac{d-a}{2c} \pm \sqrt{\left(\frac{d-a}{2c}\right)^2 + \frac{b}{c}}$, or for later use:

$cL_{1,2}=-d + a_2 \pm a_2\underbrace{\sqrt{1-\frac{ad-bc}{a_2^2}}}_{=:q}$, where $a_2=\frac{a+d}{2}$.

== Derivative at the fixpoints ==

The derivative of the fractional linear map $f$ is:

$f'(z)=\frac{ad-bz}{(cz+d)^2}$

If we plug in $z=L_1$ we get:

$f'(L_1) = \frac{ad-bz}{(a_2(1+q))^2} = \frac{ad-bz}{a_2^2}/\left(1+q\right)^2 = \frac{1-q^2}{\left(1+q\right)^2} = \frac{1-q}{1+q}$

$f'(L_2) = \frac{1+q}{1-q} = 1/f'(L_1)$.

== [[Schröder coordinate]] at the fixpoints ==

Let $\sigma_k(f(z))=f'(L_k) \sigma(z)$, $\sigma_k$ is unique up to a multiplicative constant.

Then $\sigma_1(z) = \frac{z-L_1}{z-L_2}$, $\sigma_2(z)=\frac{z-L_2}{z-L_1}$.

=== Proof ===

$\sigma_1\left(\frac{az+b}{cz+d}\right) = \frac{(a-cL_1)z + b-dL_1}{(a-cL_2)z + b -dL_2} = \frac{a_2(1-q)z+b- dL_1}{a_2(1+q)z + b -dL_2 }=\frac{1-q}{1+q}\frac{z-\frac{dL_1-b}{cL_2+d}}{z-\frac{dL_2-b}{cL_1+d}}$.

$c^2 L_1 L_2 = (-d+a_2+a_2 q)(-d+a_2 - a_2 q) = (-d+a_2)^2 - a_2^2 q^2 = \frac{(a-d)^2}{4} - \frac{(a+d)^2}{4} + ad - bc = -bc$

$(cL_2+d)L_1 = -b + dL_1$

$\sigma_1\left(\frac{az+b}{cz+d}\right) = \frac{1-q}{1+q} \frac{z-L_1}{z-L_2} = f'(L_1) \sigma_1(z)$

== Iterates ==

The iterates $f^t$ of $f$ can then be given by:

$f_k^t(z) = \sigma_k^{-1}(f'(L_k)^t \sigma_k(z))$

$\sigma_1^{-1}(w) = \frac{L_2w-L_1}{w-1}$

$f_1^t(z) = \frac{L_2K^t\frac{z-L_1}{z-L_2} - L_1}{K^t\frac{z-L_1}{z-L_2}-1} = \frac{(L_2K^t-L_1)z -L_2K^tL_1 + L_2L_1}{(K^t-1)z+K^tL_1-L_2} =
\frac{(L_2K^t-L_1)z + (K^t-1)\frac{b}{c}}{(K^t-1)z + K^tL_1 - L_2} = \frac{(L_2 (1-q)^t-L_1 (1+q)^t)z + ((1-q)^t-(1+q)^t)\frac{b}{c}}{((1-q)^t-(1+q)^t)z + (1-q)^tL_1 - (1+q)^tL_2}$

$s_t := (1-q)^t - (1+q)^t$

$f^t(z) = \frac{(-d s_t + a_2 s_{t+1})z + s_t b}{s_t c z - d s_t + a_2 (1-q^2) s_{t-1}} = \frac{(-d+a_2 \frac{s_{t+1}}{s_t})z + b}{cz-d+a_2\frac{(1-q^2)s_{t-1}}{s_t}}$

We see indeed that this is independent of the sign of $q$, i.e. regardless at which fixpoint the iterate is developed, the result is the same.

Iteration of fractional linear maps

2013-05-01T15:55:27Z

Bo198214: /* Iterates */ fixed sign error

Our concern here are functions of the form $f(z)=\frac{az+b}{cz+d}$ with $c\neq 0$.

== Fixpoints ==
In the case $c\neq 0$, the fixpoints are given by

$L_{1,2}=-\frac{d-a}{2c} \pm \sqrt{\left(\frac{d-a}{2c}\right)^2 + \frac{b}{c}}$, or for later use:

$cL_{1,2}=-d + a_2 \pm a_2\underbrace{\sqrt{1-\frac{ad-bc}{a_2^2}}}_{=:q}$, where $a_2=\frac{a+d}{2}$.

== Derivative at the fixpoints ==

The derivative of the fractional linear map $f$ is:

$f'(z)=\frac{ad-bz}{(cz+d)^2}$

If we plug in $z=L_1$ we get:

$f'(L_1) = \frac{ad-bz}{(a_2(1+q))^2} = \frac{ad-bz}{a_2^2}/\left(1+q\right)^2 = \frac{1-q^2}{\left(1+q\right)^2} = \frac{1-q}{1+q}$

$f'(L_2) = \frac{1+q}{1-q} = 1/f'(L_1)$.

== [[Schröder coordinate]] at the fixpoints ==

Let $\sigma_k(f(z))=f'(L_k) \sigma(z)$, $\sigma_k$ is unique up to a multiplicative constant.

Then $\sigma_1(z) = \frac{z-L_1}{z-L_2}$, $\sigma_2(z)=\frac{z-L_2}{z-L_1}$.

=== Proof ===

$\sigma_1\left(\frac{az+b}{cz+d}\right) = \frac{(a-cL_1)z + b-dL_1}{(a-cL_2)z + b -dL_2} = \frac{a_2(1-q)z+b- dL_1}{a_2(1+q)z + b -dL_2 }=\frac{1-q}{1+q}\frac{z-\frac{dL_1-b}{cL_2+d}}{z-\frac{dL_2-b}{cL_1+d}}$.

$c^2 L_1 L_2 = (-d+a_2+a_2 q)(-d+a_2 - a_2 q) = (-d+a_2)^2 - a_2^2 q^2 = \frac{(a-d)^2}{4} - \frac{(a+d)^2}{4} + ad - bc = -bc$

$(cL_2+d)L_1 = -b + dL_1$

$\sigma_1\left(\frac{az+b}{cz+d}\right) = \frac{1-q}{1+q} \frac{z-L_1}{z-L_2} = f'(L_1) \sigma_1(z)$

== Iterates ==

The iterates $f^t$ of $f$ can then be given by:

$f_k^t(z) = \sigma_k^{-1}(f'(L_k)^t \sigma_k(z))$

$\sigma_1^{-1}(w) = \frac{L_2w-L_1}{w-1}$

$f_1^t(z) = \frac{L_2K^t\frac{z-L_1}{z-L_2} - L_1}{K^t\frac{z-L_1}{z-L_2}-1} = \frac{(L_2K^t-L_1)z -L_2K^tL_1 + L_2L_1}{(K^t-1)z+K^tL_1-L_2} =
\frac{(L_2K^t-L_1)z + (K^t-1)\frac{b}{c}}{(K^t-1)z + K^tL_1 - L_2} = \frac{(L_2 (1-q)^t-L_1 (1+q)^t)z + ((1-q)^t-(1+q)^t)\frac{b}{c}}{((1-q)^t-(1+q)^t)z + (1-q)^tL_1 - (1+q)^tL_2}$

$s_t := (1-q)^t - (1+q)^t$

$f^t(z) = \frac{(-d s_t + a_2 s_{t+1})z + s_t b}{s_t c z - d s_t + a_2 (1-q^2) s_{t-1}} = \frac{(-d+a_2 \frac{s_{t+1}}{s_t})z + b}{cz-d+a_2(1-q^2)\frac{s_{t-1}}{s_t}}$

We see indeed that this is independent of the sign of $q$, i.e. regardless at which fixpoint the iterate is developed, the result is the same.

Iteration of fractional linear maps

2013-05-01T15:44:28Z

Bo198214: /* Iterates */

Our concern here are functions of the form $f(z)=\frac{az+b}{cz+d}$ with $c\neq 0$.

== Fixpoints ==
In the case $c\neq 0$, the fixpoints are given by

$L_{1,2}=-\frac{d-a}{2c} \pm \sqrt{\left(\frac{d-a}{2c}\right)^2 + \frac{b}{c}}$, or for later use:

$cL_{1,2}=-d + a_2 \pm a_2\underbrace{\sqrt{1-\frac{ad-bc}{a_2^2}}}_{=:q}$, where $a_2=\frac{a+d}{2}$.

== Derivative at the fixpoints ==

The derivative of the fractional linear map $f$ is:

$f'(z)=\frac{ad-bz}{(cz+d)^2}$

If we plug in $z=L_1$ we get:

$f'(L_1) = \frac{ad-bz}{(a_2(1+q))^2} = \frac{ad-bz}{a_2^2}/\left(1+q\right)^2 = \frac{1-q^2}{\left(1+q\right)^2} = \frac{1-q}{1+q}$

$f'(L_2) = \frac{1+q}{1-q} = 1/f'(L_1)$.

== [[Schröder coordinate]] at the fixpoints ==

Let $\sigma_k(f(z))=f'(L_k) \sigma(z)$, $\sigma_k$ is unique up to a multiplicative constant.

Then $\sigma_1(z) = \frac{z-L_1}{z-L_2}$, $\sigma_2(z)=\frac{z-L_2}{z-L_1}$.

=== Proof ===

$\sigma_1\left(\frac{az+b}{cz+d}\right) = \frac{(a-cL_1)z + b-dL_1}{(a-cL_2)z + b -dL_2} = \frac{a_2(1-q)z+b- dL_1}{a_2(1+q)z + b -dL_2 }=\frac{1-q}{1+q}\frac{z-\frac{dL_1-b}{cL_2+d}}{z-\frac{dL_2-b}{cL_1+d}}$.

$c^2 L_1 L_2 = (-d+a_2+a_2 q)(-d+a_2 - a_2 q) = (-d+a_2)^2 - a_2^2 q^2 = \frac{(a-d)^2}{4} - \frac{(a+d)^2}{4} + ad - bc = -bc$

$(cL_2+d)L_1 = -b + dL_1$

$\sigma_1\left(\frac{az+b}{cz+d}\right) = \frac{1-q}{1+q} \frac{z-L_1}{z-L_2} = f'(L_1) \sigma_1(z)$

== Iterates ==

The iterates $f^t$ of $f$ can then be given by:

$f_k^t(z) = \sigma_k^{-1}(f'(L_k)^t \sigma_k(z))$

$\sigma_1^{-1}(w) = \frac{L_2w-L_1}{w-1}$

$f_1^t(z) = \frac{L_2K^t\frac{z-L_1}{z-L_2} - L_1}{K^t\frac{z-L_1}{z-L_2}-1} = \frac{(L_2K^t-L_1)z -L_2K^tL_1 + L_2L_1}{(K^t-1)z-K^tL_1+L_2} =
\frac{(L_2K^t-L_1)z + (K^t-1)\frac{b}{c}}{(K^t-1)z - K^tL_1 + L_2} = \frac{(L_2 (1-q)^t-L_1 (1+q)^t)z + ((1-q)^t-(1+q)^t)\frac{b}{c}}{((1-q)^t-(1+q)^t)z - (1-q)^tL_1 + (1+q)^tL_2}$

$s_t := (1-q)^t - (1+q)^t$

$f^t(z) = \frac{(-d s_t + a_2 s_{t+1})z + s_t b}{s_t c z + d s_t - a_2 (1-q^2) s_{t-1}} = \frac{(-d+a_2 \frac{s_{t+1}}{s_t})z + b}{cz+d-a_2(1-q^2)\frac{s_{t-1}}{s_t}}$

We see indeed that this is independent of the sign of $q$, i.e. regardless at which fixpoint the iterate is developed, the result is the same.

Iteration of fractional linear maps

2013-05-01T15:23:56Z

Bo198214: /* Iterates */ finished

Our concern here are functions of the form $f(z)=\frac{az+b}{cz+d}$ with $c\neq 0$.

== Fixpoints ==
In the case $c\neq 0$, the fixpoints are given by

$L_{1,2}=-\frac{d-a}{2c} \pm \sqrt{\left(\frac{d-a}{2c}\right)^2 + \frac{b}{c}}$, or for later use:

$cL_{1,2}=-d + a_2 \pm a_2\underbrace{\sqrt{1-\frac{ad-bc}{a_2^2}}}_{=:q}$, where $a_2=\frac{a+d}{2}$.

== Derivative at the fixpoints ==

The derivative of the fractional linear map $f$ is:

$f'(z)=\frac{ad-bz}{(cz+d)^2}$

If we plug in $z=L_1$ we get:

$f'(L_1) = \frac{ad-bz}{(a_2(1+q))^2} = \frac{ad-bz}{a_2^2}/\left(1+q\right)^2 = \frac{1-q^2}{\left(1+q\right)^2} = \frac{1-q}{1+q}$

$f'(L_2) = \frac{1+q}{1-q} = 1/f'(L_1)$.

== [[Schröder coordinate]] at the fixpoints ==

Let $\sigma_k(f(z))=f'(L_k) \sigma(z)$, $\sigma_k$ is unique up to a multiplicative constant.

Then $\sigma_1(z) = \frac{z-L_1}{z-L_2}$, $\sigma_2(z)=\frac{z-L_2}{z-L_1}$.

=== Proof ===

$\sigma_1\left(\frac{az+b}{cz+d}\right) = \frac{(a-cL_1)z + b-dL_1}{(a-cL_2)z + b -dL_2} = \frac{a_2(1-q)z+b- dL_1}{a_2(1+q)z + b -dL_2 }=\frac{1-q}{1+q}\frac{z-\frac{dL_1-b}{cL_2+d}}{z-\frac{dL_2-b}{cL_1+d}}$.

$c^2 L_1 L_2 = (-d+a_2+a_2 q)(-d+a_2 - a_2 q) = (-d+a_2)^2 - a_2^2 q^2 = \frac{(a-d)^2}{4} - \frac{(a+d)^2}{4} + ad - bc = -bc$

$(cL_2+d)L_1 = -b + dL_1$

$\sigma_1\left(\frac{az+b}{cz+d}\right) = \frac{1-q}{1+q} \frac{z-L_1}{z-L_2} = f'(L_1) \sigma_1(z)$

== Iterates ==

The iterates $f^t$ of $f$ can then be given by:

$f^t(z) = \sigma_k^{-1}(f'(L_k)^t \sigma_k(z))$

$\sigma_1^{-1}(w) = \frac{L_2w-L_1}{w-1}$

$f^t(z) = \frac{L_2K^t\frac{z-L_1}{z-L_2} - L_1}{K^t\frac{z-L_1}{z-L_2}-1} = \frac{(L_2K^t-L_1)z -L_2K^tL_1 + L_2L_1}{(K^t-1)z-K^tL_1+L_2} =
\frac{(L_2K^t-L_1)z + (K^t-1)\frac{b}{c}}{(K^t-1)z - K^tL_1 + L_2} = \frac{(L_2 (1-q)^t-L_1 (1+q)^t)z + ((1-q)^t-(1+q)^t)\frac{b}{c}}{((1-q)^t-(1+q)^t)z - (1-q)^tL_1 + (1+q)^tL_2}$

$s_t := (1-q)^t - (1+q)^t$

$f^t(z) = \frac{(-d s_t + a_2 s_{t+1})z + s_t b}{s_t c z + d s_t - a_2 (1-q^2) s_{t-1}} = \frac{(-d+a_2 \frac{s_{t+1}}{s_t})z + b}{cz+d-a_2(1-q^2)\frac{s_{t-1}}{s_t}}$

We see indeed that this is independent of the sign of $q$, i.e. regardless at which fixpoint the iterate is developed, the result is the same.

Iteration of fractional linear maps

2013-05-01T14:47:17Z

Bo198214: /* Iterates */ intermediate

Our concern here are functions of the form $f(z)=\frac{az+b}{cz+d}$ with $c\neq 0$.

== Fixpoints ==
In the case $c\neq 0$, the fixpoints are given by

$L_{1,2}=-\frac{d-a}{2c} \pm \sqrt{\left(\frac{d-a}{2c}\right)^2 + \frac{b}{c}}$, or for later use:

$cL_{1,2}=-d + a_2 \pm a_2\underbrace{\sqrt{1-\frac{ad-bc}{a_2^2}}}_{=:q}$, where $a_2=\frac{a+d}{2}$.

== Derivative at the fixpoints ==

The derivative of the fractional linear map $f$ is:

$f'(z)=\frac{ad-bz}{(cz+d)^2}$

If we plug in $z=L_1$ we get:

$f'(L_1) = \frac{ad-bz}{(a_2(1+q))^2} = \frac{ad-bz}{a_2^2}/\left(1+q\right)^2 = \frac{1-q^2}{\left(1+q\right)^2} = \frac{1-q}{1+q}$

$f'(L_2) = \frac{1+q}{1-q} = 1/f'(L_1)$.

== [[Schröder coordinate]] at the fixpoints ==

Let $\sigma_k(f(z))=f'(L_k) \sigma(z)$, $\sigma_k$ is unique up to a multiplicative constant.

Then $\sigma_1(z) = \frac{z-L_1}{z-L_2}$, $\sigma_2(z)=\frac{z-L_2}{z-L_1}$.

=== Proof ===

$\sigma_1\left(\frac{az+b}{cz+d}\right) = \frac{(a-cL_1)z + b-dL_1}{(a-cL_2)z + b -dL_2} = \frac{a_2(1-q)z+b- dL_1}{a_2(1+q)z + b -dL_2 }=\frac{1-q}{1+q}\frac{z-\frac{dL_1-b}{cL_2+d}}{z-\frac{dL_2-b}{cL_1+d}}$.

$c^2 L_1 L_2 = (-d+a_2+a_2 q)(-d+a_2 - a_2 q) = (-d+a_2)^2 - a_2^2 q^2 = \frac{(a-d)^2}{4} - \frac{(a+d)^2}{4} + ad - bc = -bc$

$(cL_2+d)L_1 = -b + dL_1$

$\sigma_1\left(\frac{az+b}{cz+d}\right) = \frac{1-q}{1+q} \frac{z-L_1}{z-L_2} = f'(L_1) \sigma_1(z)$

== Iterates ==

The iterates $f^t$ of $f$ can then be given by:

$f^t(z) = \sigma_k^{-1}(f'(L_k)^t \sigma_k(z))$

$\sigma_1^{-1}(w) = \frac{L_2w-L_1}{w-1}$

$f^t(z) = \frac{L_2K^t\frac{z-L_1}{z-L_2} - L_1}{K^t\frac{z-L_1}{z-L_2}-1} = \frac{(L_2K^t-L_1)z -L_2K^tL_1 + L_2L_1}{(K^t-1)z-K^tL_1+L_2} =
\frac{(L_2K^t-L_1)z + (K^t-1)\frac{b}{c}}{(K^t-1)z - K^tL_1 + L_2} = \frac{(L_2 (1-q)^t-L_1 (1+q)^t)z + ((1-q)^t-(1+q)^t)\frac{b}{c}}{((1-q)^t-(1+q)^t)z - (1-q)^tL_1 + (1+q)^tL_2}$

Iteration of fractional linear maps

2013-05-01T14:31:55Z

Bo198214: /* Schröder coordinate at the fixpoints */ save state

Our concern here are functions of the form $f(z)=\frac{az+b}{cz+d}$ with $c\neq 0$.

== Fixpoints ==
In the case $c\neq 0$, the fixpoints are given by

$L_{1,2}=-\frac{d-a}{2c} \pm \sqrt{\left(\frac{d-a}{2c}\right)^2 + \frac{b}{c}}$, or for later use:

$cL_{1,2}=-d + a_2 \pm a_2\underbrace{\sqrt{1-\frac{ad-bc}{a_2^2}}}_{=:q}$, where $a_2=\frac{a+d}{2}$.

== Derivative at the fixpoints ==

The derivative of the fractional linear map $f$ is:

$f'(z)=\frac{ad-bz}{(cz+d)^2}$

If we plug in $z=L_1$ we get:

$f'(L_1) = \frac{ad-bz}{(a_2(1+q))^2} = \frac{ad-bz}{a_2^2}/\left(1+q\right)^2 = \frac{1-q^2}{\left(1+q\right)^2} = \frac{1-q}{1+q}$

$f'(L_2) = \frac{1+q}{1-q} = 1/f'(L_1)$.

== [[Schröder coordinate]] at the fixpoints ==

Let $\sigma_k(f(z))=f'(L_k) \sigma(z)$, $\sigma_k$ is unique up to a multiplicative constant.

Then $\sigma_1(z) = \frac{z-L_1}{z-L_2}$, $\sigma_2(z)=\frac{z-L_2}{z-L_1}$.

=== Proof ===

$\sigma_1\left(\frac{az+b}{cz+d}\right) = \frac{(a-cL_1)z + b-dL_1}{(a-cL_2)z + b -dL_2} = \frac{a_2(1-q)z+b- dL_1}{a_2(1+q)z + b -dL_2 }=\frac{1-q}{1+q}\frac{z-\frac{dL_1-b}{cL_2+d}}{z-\frac{dL_2-b}{cL_1+d}}$.

$c^2 L_1 L_2 = (-d+a_2+a_2 q)(-d+a_2 - a_2 q) = (-d+a_2)^2 - a_2^2 q^2 = \frac{(a-d)^2}{4} - \frac{(a+d)^2}{4} + ad - bc = -bc$

$(cL_2+d)L_1 = -b + dL_1$

$\sigma_1\left(\frac{az+b}{cz+d}\right) = \frac{1-q}{1+q} \frac{z-L_1}{z-L_2} = f'(L_1) \sigma_1(z)$

== Iterates ==

The iterates $f^t$ of $f$ can then be given by:

$f^t(z) = \sigma_k^{-1}(f'(L_k)^t \sigma_k(z))$

$\sigma_1^{-1}(w) = \frac{L_2w-L_1}{w-1}$

$f^t(z) = \frac{L_2K^t\frac{z-L_1}{z-L_2} - L_1}{K^t\frac{z-L_1}{z-L_2}-1} = \frac{(L_2K^t-L_1)z -L_2K^tL_1 + L_2L_1}{(K^t-1)z-K^tL_1+L_2} = $

Iteration of fractional linear maps

2013-05-01T13:35:30Z

Bo198214: /* Fixpoints */

Iteration of fractional linear maps

2013-05-01T13:34:10Z

Bo198214: /* Fixpoints */

Iteration of fractional linear maps

2013-05-01T13:33:16Z

Bo198214: /* Fixpoints */ elegance

Iteration of fractional linear maps

2013-05-01T13:32:26Z

Bo198214: /* Fixpoints */ fixed typo

Iteration of fractional linear maps

2013-05-01T13:30:47Z

Bo198214: /* Fixpoints */

Iteration of fractional linear maps

2013-05-01T13:22:03Z

Bo198214: /* Derivative at the fixpoints */

Iteration of fractional linear maps

2013-05-01T13:20:38Z

Bo198214: /* Fixpoints */ fixed error

Iteration of fractional linear maps

2013-05-01T13:19:28Z

Bo198214: /* Fixpoints */ more poignant formula

Iteration of fractional linear maps

2013-05-01T13:14:40Z

Bo198214: /* Schröder coordinate at the fixpoints */ started derivation

Iteration of fractional linear maps

2013-05-01T12:33:44Z

Bo198214: added schroeder stub

Iteration of fractional linear maps

2013-05-01T12:27:04Z

Bo198214: /* Derivative at the fixpoints */

Iteration of fractional linear maps

2013-05-01T12:26:24Z

Bo198214: added c neq 0 condition

Iteration of fractional linear maps

2013-05-01T12:24:30Z

Bo198214: finished with derivative formula at fixpoint

Our concern here are functions of the form $f(z)=\frac{az+b}{cz+d}$.

== Fixpoints ==
In the case $c\neq 0$, the fixpoints are given by

$L_{1,2}=-\frac{d-a}{2c} \pm \sqrt{\left(\frac{d-a}{2c}\right)^2 + \frac{b}{c}}$, or for later use:

$L_{1,2}=\frac{a_1}{c} \pm \frac{a_2}{c} \underbrace{\sqrt{1-\frac{ad-bc}{a_2^2}}}_{=:q}$, where $a_1=\frac{a-d}{2}$ and $a_2=\frac{a+d}{2}$.

== Derivative at the fixpoints ==

The derivative of the fractional linear map $f$ is:

$f'(z)=\frac{ad-bz}{(cz+d)^2}$

If we plug in $z=L_1$ we get:

$f'(L_1) = \frac{ad-bz}{(a_1+a_2 q + d)^2} = \frac{ad-bz}{a_2^2}/\left(1+q\right)^2 = \frac{1-q^2}{\left(1+q\right)^2} = \frac{1-q}{1+q}$

$f'(L_2) = \frac{1+q}{1-q}$.

Fixpoint

2013-05-01T11:57:01Z

Bo198214: added example

A fixpoint $p$ of a function $f$ is a value such that $f(p)=p$.

If $f$ is a holomorphic function, we define the multiplicity of the fixpoint $p$ to be the [http://en.wikipedia.org/wiki/Zero_(complex_analysis) multiplicity of the zero] of $f(z)-z$ at $p$.

For example $z+z^2$ has multiplicity 2 at fixpoint 0, $2z+z^2$ has multiplicity 1 at fixpoint 0.

Fixpoint

2013-05-01T11:53:20Z

Bo198214: linkt to wikipedia

Fixpoint

2013-05-01T11:51:13Z

Bo198214: Added multiplicity.

A fixpoint $p$ of a function $f$ is a value such that $f(p)=p$.

If $f$ is a holomorphic function, we define the multiplicity of the fixpoint $p$ to be the multiplicity of the zero of $f(z)-z$ at $p$.

Shishikura perturbed Fatou coordinates

2013-02-19T21:52:44Z

Bo198214: /* Proposition 3.2.2 */ used wiki numbering

Shishikura writes<ref name="Shishikura2000">Shishikura, M. (2000). Bifurcation of parabolic fixed points. In Lei, Tan, The Mandelbrot set, theme and variations. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 274, 325-363 (2000)</ref> the following

== $\mathcal{F}_0$ ==
[p. 327]

If $f_0''(0)\neq 0$ by another coordinate change we may assume that $f_0''(0)=1$, so define
$$ \mathcal{F}_0 = \{ f_0 \colon f_0 \quad\text{is defined and analytic in a neighborhood of 0}\quad f_0(0)=0, f_0'(0)=1, f_0''(0)=1\}$$

== Neighborhood of $f$ in the compact-open topology with domain of definition ==
[ p. 332 ]

In what follows a map $f$ is always associated with its domain of definition $D(f)$. That is two maps are considered as distinct if they have distinct domains, even if the one is an extension of the other. A neighborhood of an analytic map $f\colon D(f)\to\overline{\C}$ is a set containing
$$\{ g \colon g \quad\text{is analytic},\quad D(g)\supset K, \sup_{z\in K} d(f(z)-g(z))<\eps \} $$
where $K$ is a compact set in $D(f)$, $\eps>0$ and $d(.,.)$ is the spherical metric.
A sequence $(f_n)$ converges to $f$ in this topology if and only if for any compact set $K\subset D(f)$ there exists $n_0$ such that $K\subset D(f_n)$ for all $n\ge n_0$ and $f_n|_K\to f|_K$ uniformly on $K$ as $n\to\infty$. The system of these neighborhoods define the "compact-open topology together with the domain of defintion", which is unfortunately not Hausdorff, since an extension of $f$ is contained in any neighborhood of $f$.

For $b_1,b_2\in\C$ with $\Re(b_1)<\Re(b_2)$ define
$$Q(b_1,b_2)=\{z\in\C\colon \Re(z-b_1)>-|\Im(z-b_1)|, \Re(z-b_2)>|\Im(z-b_2)|\}$$
If $b_1=-\infty$ (resp. $b_2=\infty$) then the corresponding condition should be removed.

== Proposition 2.5.2 ==
[p. 333]

Let $f$ be a holomorphic function defined on $Q=Q(b_1,b_2)$ with $\Re(b_2)>\Re(b_1)+2$ (here $b_1$ or $b_2$ may be $-\infty$ or $\infty$). Suppose
$$|F(z)-(z+1)|<\frac{1}{4},\quad |F'(z)-(z+1)|<\frac{1}{4}\quad \text{for all}\quad z\in Q.$$

Then
# $F$ is univalent [means injective] on $Q$.
# Let $z_0\in Q$ be a point such that $\Re(b_1)<\Re(z_0)<\Re(b_2)-\frac{5}{4}$. Denote by $S$ the closed region (a strip) bounded by the two curves $\ell=\{z_0+i y\colon y\in\R\}$ and $F(\ell)$. Then for any $z\in Q$ there exist a unique $n\in \Z$ such that $F^n(z)$ is defined and belongs to $S-F(\ell)$.
# There exists an univalent function $\Phi\colon Q\to \C$ satisfying $$\Phi(F(z))=\Phi(z)+1$$ whenever both sides are defined. Moreover $\Phi$ is unique up to an addition of a constant.
# Fix a point $z_0\in Q$. If we normalize $\Phi$ by $\Phi(z_0)=0$ then the correspondence $F\mapsto \Phi$ is continuous with respect to the compact-open topology.

== $\mathcal{F}$, $\mathcal{F}_1$ and $\alpha(f)$ ==
[p. 339]

$$ \mathcal{F} = \{ f \colon f \quad\text{is defined and analytic in a neighborhood of 0}\quad f(0)=0, f_0'(0)\neq 0\}$$
For $f\in \mathcal{F}$ we express the derivative
$$f'(0)=\exp(2\pi i \alpha(f))$$
where $\alpha(f)\in\C$ and $-\frac{1}{2} < \Re(\alpha(f)) \le \frac{1}{2}$ our study is focussed on the maps in the following class
$$ \mathcal{F}_1 = \{ f\in \mathcal{F}\colon |\arg(\alpha(f))| < \frac{\pi}{4}\} $$

=== Comments ===

By the definition we have $$
\begin{align*}
\log(f'(0))&=2\pi i\alpha(f)=2\pi i re^{i\beta}=2\pi r e^{i(\beta+\frac{\pi}{2})}\\
\arg(\log(f'(0))&=\beta+\frac{\pi}{2}
\end{align*}$$
so the condition for $\mathcal{F}_1$ is
$|\arg(\log(f'(0)))-\frac{\pi}{2}|<\frac{\pi}{4}$ or
$$\frac{\pi}{4} < \arg(\log(f'(0)) < \frac{3\pi}{4}$$

== $\sigma(f)$ ==
[p. 340]

Let $\sigma(f)$ be the other fixpoint of $f$ (besides 0).

...

$$\sigma(f) = 2\pi i \alpha(f)(1+o(1)), \quad\text{as}\quad f\to f_0$$

=== Comments ===

By the comment before, we can say that $\sigma(f)$ lies roughly in the sector of angle $(\frac{\pi}{4},\frac{3\pi}{4})$ above the tip 0.

== Proposition 3.2.2 ==
[p. 340]

Let $f_0\in\mathcal{F}_0$. There exists a neighborhood $\mathcal{N}_0$ of $f_0$ (in the sense of the topology defined in 2.5.1) such that if $f\in\mathcal{N}_0\cap \mathcal{F}_1$, then there exist closed Jordan domains $S_{-,f}$, $S_{+,f}$ and analytic functions $\Phi_{-,f}$, $\Phi_{+,f}$ satisfying the following:

# $S_{+,f}$ is bounded by an arc $\ell_{\pm}$ and its image $f(\ell_{\pm})$ such that $\ell_{\pm}$ joins two fixed points $\{0,\sigma(f)\}$, $\ell_{\pm}\cap f(\ell_{\pm}) = \{0,\sigma(f)\}$ and $S_{-,f}\cap S_{+,f} = \{0,\sigma(f)\}$;
# $\Phi_{\pm,f}$ is defined, analytic and injective in a neighborhood of $S'_{\pm,f} = S_{\pm,f}\setminus \{0,\sigma(f)\}$;
# if $z\in S'_{-,f}$ then there is an $n\ge 1$ such that $f^n(z)\in S'_{+,f}$ and for the smallest such $n$, \[ \Phi_{+,f}(f^n(z)) = \Phi_{-,f}(z) - \frac{1}{\alpha(f)} + n \]
# when $f\in\mathcal{F}_1$ tends to $f_0$, $S_{\pm,f}\to S_{\pm,0}\cup \{0\}$ in the Hausdorff metric, and $\Phi_{\pm,f}\to \Phi_{\pm,0}$ in the topology defined in 2.5.1.

== Proposition 4.4.1 ==
[p. 356]

Let $f_0\in\mathcal{F}_0$, there exists a neighborhood $\mathcal{N}_0$ of $f_0$ (in the previously described "compact-open topology with domain of definition"), such that if $f\in \mathcal{N}_0 \cap \mathcal{F}_1$, then there exist large $\xi_0, \eta_0>0$, and analytic maps $\ph_f\colon D(\ph_f)\to\overline{\C}$ and $\tilde{\mathcal{R}}_f\colon D(\tilde{\mathcal{R}}_f)\to \C$ satisfying the following:
# $D(\ph_f)$ contains the set $$ \mathcal{Q}_f = \{ w\in\C\colon |\arg(-w-\xi_0)| < \frac{2\pi}{3}, |\arg(w+\frac{1}{\alpha(f)}-\xi_0)|<\frac{2\pi}{3} \} $$ and $\ph_f(D(\ph_f)) \subseteq D(f)$; If $w, w+1\in D(\ph_f)$ then $$ \ph_f(w+1)=f(\ph_f(w)) $$ $\ph_f(w)\to 0$ when $w\in \mathcal{Q}_f$ and $\Im(w)\to\infty$; $\ph_f(w)\to\sigma(f)$ when $w\in\mathcal{Q}_f$ and $\Im(w)\to -\infty$.
# $D(\tilde{\mathcal{R}}_f)$ contains $\{ w\in\C\colon |\Im(w)| > \eta_0\}$ and is invariant under $T(w)=w+1$; if $w\in D(\tilde{\mathcal{R}}_f)$ then $$\tilde{\mathcal{R}}_f(w+1)=\tilde{\mathcal{R}}(w)+1;$$ $\tilde{\mathcal{R}}_f(w)-w$ tends to $-\frac{1}{\alpha(f)}$ when $\Im(w)\to\infty$; to a constant when $\Im(w)\to -\infty$.
# If $w\in D(\tilde{\mathcal{R}}_f)\cap D(\ph_f)$ and $w'=\tilde{\mathcal{R}}_f(w)+n\in D(\ph_f)$ for some $n\in\Z$ then $$f^n(\ph_f(w))=\ph_f(w')\quad \text{for}\quad n\ge 0, \quad f^{-n}(\ph_f(w'))=\ph_f(w)\quad \text{for}\quad n<0$$ Moreover if $|\arg(w'+\frac{1}{2\alpha(f)}-\xi_0)|<\frac{2\pi}{3}$ then $n>0$.
# When $f\in\mathcal{F_1}$ and $f\to f_0$ $$\ph_f\to\ph_0\quad\text{and}\quad \tilde{\mathcal{R}}_f + \frac{1}{\alpha(f)} \to \tilde{\mathcal{E}}_{f_0}$$

== Proposition A.1 ==
[p. 360]

In the settings of Proposition 3.2.2 (resp. 2.5.2), assume in addtion that $f_s(z)$ (resp. $F_s(z)$) is a family of holomorphic maps for $s\in \Delta \cup \{0\} \subset \mathbb{C}^k$, with $\Delta\subset\mathbb{C}^k$ an open connected set containing 0 in the boundary, such that $f_s$ (resp. $F_s(z)$) is continuous for $s\in\Delta\cup \{0\}$ and analytic in $s\in\Delta$, then the Fatou coordinates $\Phi_{\pm,f_s}$ (resp. $\Phi_{F_s}$) depend analytically on $s$ for $s\in\Delta$.

<references/>

Shishikura perturbed Fatou coordinates

2013-02-19T21:50:33Z

Bo198214: /* Proposition A.1 */ added page

Shishikura writes<ref name="Shishikura2000">Shishikura, M. (2000). Bifurcation of parabolic fixed points. In Lei, Tan, The Mandelbrot set, theme and variations. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 274, 325-363 (2000)</ref> the following

== $\mathcal{F}_0$ ==
[p. 327]

If $f_0''(0)\neq 0$ by another coordinate change we may assume that $f_0''(0)=1$, so define
$$ \mathcal{F}_0 = \{ f_0 \colon f_0 \quad\text{is defined and analytic in a neighborhood of 0}\quad f_0(0)=0, f_0'(0)=1, f_0''(0)=1\}$$

== Neighborhood of $f$ in the compact-open topology with domain of definition ==
[ p. 332 ]

In what follows a map $f$ is always associated with its domain of definition $D(f)$. That is two maps are considered as distinct if they have distinct domains, even if the one is an extension of the other. A neighborhood of an analytic map $f\colon D(f)\to\overline{\C}$ is a set containing
$$\{ g \colon g \quad\text{is analytic},\quad D(g)\supset K, \sup_{z\in K} d(f(z)-g(z))<\eps \} $$
where $K$ is a compact set in $D(f)$, $\eps>0$ and $d(.,.)$ is the spherical metric.
A sequence $(f_n)$ converges to $f$ in this topology if and only if for any compact set $K\subset D(f)$ there exists $n_0$ such that $K\subset D(f_n)$ for all $n\ge n_0$ and $f_n|_K\to f|_K$ uniformly on $K$ as $n\to\infty$. The system of these neighborhoods define the "compact-open topology together with the domain of defintion", which is unfortunately not Hausdorff, since an extension of $f$ is contained in any neighborhood of $f$.

For $b_1,b_2\in\C$ with $\Re(b_1)<\Re(b_2)$ define
$$Q(b_1,b_2)=\{z\in\C\colon \Re(z-b_1)>-|\Im(z-b_1)|, \Re(z-b_2)>|\Im(z-b_2)|\}$$
If $b_1=-\infty$ (resp. $b_2=\infty$) then the corresponding condition should be removed.

== Proposition 2.5.2 ==
[p. 333]

Let $f$ be a holomorphic function defined on $Q=Q(b_1,b_2)$ with $\Re(b_2)>\Re(b_1)+2$ (here $b_1$ or $b_2$ may be $-\infty$ or $\infty$). Suppose
$$|F(z)-(z+1)|<\frac{1}{4},\quad |F'(z)-(z+1)|<\frac{1}{4}\quad \text{for all}\quad z\in Q.$$

Then
# $F$ is univalent [means injective] on $Q$.
# Let $z_0\in Q$ be a point such that $\Re(b_1)<\Re(z_0)<\Re(b_2)-\frac{5}{4}$. Denote by $S$ the closed region (a strip) bounded by the two curves $\ell=\{z_0+i y\colon y\in\R\}$ and $F(\ell)$. Then for any $z\in Q$ there exist a unique $n\in \Z$ such that $F^n(z)$ is defined and belongs to $S-F(\ell)$.
# There exists an univalent function $\Phi\colon Q\to \C$ satisfying $$\Phi(F(z))=\Phi(z)+1$$ whenever both sides are defined. Moreover $\Phi$ is unique up to an addition of a constant.
# Fix a point $z_0\in Q$. If we normalize $\Phi$ by $\Phi(z_0)=0$ then the correspondence $F\mapsto \Phi$ is continuous with respect to the compact-open topology.

== $\mathcal{F}$, $\mathcal{F}_1$ and $\alpha(f)$ ==
[p. 339]

$$ \mathcal{F} = \{ f \colon f \quad\text{is defined and analytic in a neighborhood of 0}\quad f(0)=0, f_0'(0)\neq 0\}$$
For $f\in \mathcal{F}$ we express the derivative
$$f'(0)=\exp(2\pi i \alpha(f))$$
where $\alpha(f)\in\C$ and $-\frac{1}{2} < \Re(\alpha(f)) \le \frac{1}{2}$ our study is focussed on the maps in the following class
$$ \mathcal{F}_1 = \{ f\in \mathcal{F}\colon |\arg(\alpha(f))| < \frac{\pi}{4}\} $$

=== Comments ===

By the definition we have $$
\begin{align*}
\log(f'(0))&=2\pi i\alpha(f)=2\pi i re^{i\beta}=2\pi r e^{i(\beta+\frac{\pi}{2})}\\
\arg(\log(f'(0))&=\beta+\frac{\pi}{2}
\end{align*}$$
so the condition for $\mathcal{F}_1$ is
$|\arg(\log(f'(0)))-\frac{\pi}{2}|<\frac{\pi}{4}$ or
$$\frac{\pi}{4} < \arg(\log(f'(0)) < \frac{3\pi}{4}$$

== $\sigma(f)$ ==
[p. 340]

Let $\sigma(f)$ be the other fixpoint of $f$ (besides 0).

...

$$\sigma(f) = 2\pi i \alpha(f)(1+o(1)), \quad\text{as}\quad f\to f_0$$

=== Comments ===

By the comment before, we can say that $\sigma(f)$ lies roughly in the sector of angle $(\frac{\pi}{4},\frac{3\pi}{4})$ above the tip 0.

== Proposition 3.2.2 ==
[p. 340]

Let $f_0\in\mathcal{F}_0$. There exists a neighborhood $\mathcal{N}_0$ of $f_0$ (in the sense of the topology defined in 2.5.1) such that if $f\in\mathcal{N}_0\cap \mathcal{F}_1$, then there exist closed Jordan domains $S_{-,f}$, $S_{+,f}$ and analytic functions $\Phi_{-,f}$, $\Phi_{+,f}$ satisfying the following:

(i) $S_{+,f}$ is bounded by an arc $\ell_{\pm}$ and its image $f(\ell_{\pm})$ such that $\ell_{\pm}$ joins two fixed points $\{0,\sigma(f)\}$, $\ell_{\pm}\cap f(\ell_{\pm}) = \{0,\sigma(f)\}$ and $S_{-,f}\cap S_{+,f} = \{0,\sigma(f)\}$;

(ii) $\Phi_{\pm,f}$ is defined, analytic and injective in a neighborhood of $S'_{\pm,f} = S_{\pm,f}\setminus \{0,\sigma(f)\}$;

(iii) if $z\in S'_{-,f}$ then there is an $n\ge 1$ such that $f^n(z)\in S'_{+,f}$ and for the smallest such $n$,
\[ \Phi_{+,f}(f^n(z)) = \Phi_{-,f}(z) - \frac{1}{\alpha(f)} + n \]

(iv) when $f\in\mathcal{F}_1$ tends to $f_0$, $S_{\pm,f}\to S_{\pm,0}\cup \{0\}$ in the Hausdorff metric, and $\Phi_{\pm,f}\to \Phi_{\pm,0}$ in the topology defined in 2.5.1.

== Proposition 4.4.1 ==
[p. 356]

Let $f_0\in\mathcal{F}_0$, there exists a neighborhood $\mathcal{N}_0$ of $f_0$ (in the previously described "compact-open topology with domain of definition"), such that if $f\in \mathcal{N}_0 \cap \mathcal{F}_1$, then there exist large $\xi_0, \eta_0>0$, and analytic maps $\ph_f\colon D(\ph_f)\to\overline{\C}$ and $\tilde{\mathcal{R}}_f\colon D(\tilde{\mathcal{R}}_f)\to \C$ satisfying the following:
# $D(\ph_f)$ contains the set $$ \mathcal{Q}_f = \{ w\in\C\colon |\arg(-w-\xi_0)| < \frac{2\pi}{3}, |\arg(w+\frac{1}{\alpha(f)}-\xi_0)|<\frac{2\pi}{3} \} $$ and $\ph_f(D(\ph_f)) \subseteq D(f)$; If $w, w+1\in D(\ph_f)$ then $$ \ph_f(w+1)=f(\ph_f(w)) $$ $\ph_f(w)\to 0$ when $w\in \mathcal{Q}_f$ and $\Im(w)\to\infty$; $\ph_f(w)\to\sigma(f)$ when $w\in\mathcal{Q}_f$ and $\Im(w)\to -\infty$.
# $D(\tilde{\mathcal{R}}_f)$ contains $\{ w\in\C\colon |\Im(w)| > \eta_0\}$ and is invariant under $T(w)=w+1$; if $w\in D(\tilde{\mathcal{R}}_f)$ then $$\tilde{\mathcal{R}}_f(w+1)=\tilde{\mathcal{R}}(w)+1;$$ $\tilde{\mathcal{R}}_f(w)-w$ tends to $-\frac{1}{\alpha(f)}$ when $\Im(w)\to\infty$; to a constant when $\Im(w)\to -\infty$.
# If $w\in D(\tilde{\mathcal{R}}_f)\cap D(\ph_f)$ and $w'=\tilde{\mathcal{R}}_f(w)+n\in D(\ph_f)$ for some $n\in\Z$ then $$f^n(\ph_f(w))=\ph_f(w')\quad \text{for}\quad n\ge 0, \quad f^{-n}(\ph_f(w'))=\ph_f(w)\quad \text{for}\quad n<0$$ Moreover if $|\arg(w'+\frac{1}{2\alpha(f)}-\xi_0)|<\frac{2\pi}{3}$ then $n>0$.
# When $f\in\mathcal{F_1}$ and $f\to f_0$ $$\ph_f\to\ph_0\quad\text{and}\quad \tilde{\mathcal{R}}_f + \frac{1}{\alpha(f)} \to \tilde{\mathcal{E}}_{f_0}$$

== Proposition A.1 ==
[p. 360]

In the settings of Proposition 3.2.2 (resp. 2.5.2), assume in addtion that $f_s(z)$ (resp. $F_s(z)$) is a family of holomorphic maps for $s\in \Delta \cup \{0\} \subset \mathbb{C}^k$, with $\Delta\subset\mathbb{C}^k$ an open connected set containing 0 in the boundary, such that $f_s$ (resp. $F_s(z)$) is continuous for $s\in\Delta\cup \{0\}$ and analytic in $s\in\Delta$, then the Fatou coordinates $\Phi_{\pm,f_s}$ (resp. $\Phi_{F_s}$) depend analytically on $s$ for $s\in\Delta$.

<references/>

Shishikura perturbed Fatou coordinates

2013-02-19T21:49:26Z

Bo198214: /* Proposition A.1 */ fixed typo

Shishikura writes<ref name="Shishikura2000">Shishikura, M. (2000). Bifurcation of parabolic fixed points. In Lei, Tan, The Mandelbrot set, theme and variations. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 274, 325-363 (2000)</ref> the following

== $\mathcal{F}_0$ ==
[p. 327]

If $f_0''(0)\neq 0$ by another coordinate change we may assume that $f_0''(0)=1$, so define
$$ \mathcal{F}_0 = \{ f_0 \colon f_0 \quad\text{is defined and analytic in a neighborhood of 0}\quad f_0(0)=0, f_0'(0)=1, f_0''(0)=1\}$$

== Neighborhood of $f$ in the compact-open topology with domain of definition ==
[ p. 332 ]

In what follows a map $f$ is always associated with its domain of definition $D(f)$. That is two maps are considered as distinct if they have distinct domains, even if the one is an extension of the other. A neighborhood of an analytic map $f\colon D(f)\to\overline{\C}$ is a set containing
$$\{ g \colon g \quad\text{is analytic},\quad D(g)\supset K, \sup_{z\in K} d(f(z)-g(z))<\eps \} $$
where $K$ is a compact set in $D(f)$, $\eps>0$ and $d(.,.)$ is the spherical metric.
A sequence $(f_n)$ converges to $f$ in this topology if and only if for any compact set $K\subset D(f)$ there exists $n_0$ such that $K\subset D(f_n)$ for all $n\ge n_0$ and $f_n|_K\to f|_K$ uniformly on $K$ as $n\to\infty$. The system of these neighborhoods define the "compact-open topology together with the domain of defintion", which is unfortunately not Hausdorff, since an extension of $f$ is contained in any neighborhood of $f$.

For $b_1,b_2\in\C$ with $\Re(b_1)<\Re(b_2)$ define
$$Q(b_1,b_2)=\{z\in\C\colon \Re(z-b_1)>-|\Im(z-b_1)|, \Re(z-b_2)>|\Im(z-b_2)|\}$$
If $b_1=-\infty$ (resp. $b_2=\infty$) then the corresponding condition should be removed.

== Proposition 2.5.2 ==
[p. 333]

Let $f$ be a holomorphic function defined on $Q=Q(b_1,b_2)$ with $\Re(b_2)>\Re(b_1)+2$ (here $b_1$ or $b_2$ may be $-\infty$ or $\infty$). Suppose
$$|F(z)-(z+1)|<\frac{1}{4},\quad |F'(z)-(z+1)|<\frac{1}{4}\quad \text{for all}\quad z\in Q.$$

Then
# $F$ is univalent [means injective] on $Q$.
# Let $z_0\in Q$ be a point such that $\Re(b_1)<\Re(z_0)<\Re(b_2)-\frac{5}{4}$. Denote by $S$ the closed region (a strip) bounded by the two curves $\ell=\{z_0+i y\colon y\in\R\}$ and $F(\ell)$. Then for any $z\in Q$ there exist a unique $n\in \Z$ such that $F^n(z)$ is defined and belongs to $S-F(\ell)$.
# There exists an univalent function $\Phi\colon Q\to \C$ satisfying $$\Phi(F(z))=\Phi(z)+1$$ whenever both sides are defined. Moreover $\Phi$ is unique up to an addition of a constant.
# Fix a point $z_0\in Q$. If we normalize $\Phi$ by $\Phi(z_0)=0$ then the correspondence $F\mapsto \Phi$ is continuous with respect to the compact-open topology.

== $\mathcal{F}$, $\mathcal{F}_1$ and $\alpha(f)$ ==
[p. 339]

$$ \mathcal{F} = \{ f \colon f \quad\text{is defined and analytic in a neighborhood of 0}\quad f(0)=0, f_0'(0)\neq 0\}$$
For $f\in \mathcal{F}$ we express the derivative
$$f'(0)=\exp(2\pi i \alpha(f))$$
where $\alpha(f)\in\C$ and $-\frac{1}{2} < \Re(\alpha(f)) \le \frac{1}{2}$ our study is focussed on the maps in the following class
$$ \mathcal{F}_1 = \{ f\in \mathcal{F}\colon |\arg(\alpha(f))| < \frac{\pi}{4}\} $$

=== Comments ===

By the definition we have $$
\begin{align*}
\log(f'(0))&=2\pi i\alpha(f)=2\pi i re^{i\beta}=2\pi r e^{i(\beta+\frac{\pi}{2})}\\
\arg(\log(f'(0))&=\beta+\frac{\pi}{2}
\end{align*}$$
so the condition for $\mathcal{F}_1$ is
$|\arg(\log(f'(0)))-\frac{\pi}{2}|<\frac{\pi}{4}$ or
$$\frac{\pi}{4} < \arg(\log(f'(0)) < \frac{3\pi}{4}$$

== $\sigma(f)$ ==
[p. 340]

Let $\sigma(f)$ be the other fixpoint of $f$ (besides 0).

...

$$\sigma(f) = 2\pi i \alpha(f)(1+o(1)), \quad\text{as}\quad f\to f_0$$

=== Comments ===

By the comment before, we can say that $\sigma(f)$ lies roughly in the sector of angle $(\frac{\pi}{4},\frac{3\pi}{4})$ above the tip 0.

== Proposition 3.2.2 ==
[p. 340]

Let $f_0\in\mathcal{F}_0$. There exists a neighborhood $\mathcal{N}_0$ of $f_0$ (in the sense of the topology defined in 2.5.1) such that if $f\in\mathcal{N}_0\cap \mathcal{F}_1$, then there exist closed Jordan domains $S_{-,f}$, $S_{+,f}$ and analytic functions $\Phi_{-,f}$, $\Phi_{+,f}$ satisfying the following:

(i) $S_{+,f}$ is bounded by an arc $\ell_{\pm}$ and its image $f(\ell_{\pm})$ such that $\ell_{\pm}$ joins two fixed points $\{0,\sigma(f)\}$, $\ell_{\pm}\cap f(\ell_{\pm}) = \{0,\sigma(f)\}$ and $S_{-,f}\cap S_{+,f} = \{0,\sigma(f)\}$;

(ii) $\Phi_{\pm,f}$ is defined, analytic and injective in a neighborhood of $S'_{\pm,f} = S_{\pm,f}\setminus \{0,\sigma(f)\}$;

(iii) if $z\in S'_{-,f}$ then there is an $n\ge 1$ such that $f^n(z)\in S'_{+,f}$ and for the smallest such $n$,
\[ \Phi_{+,f}(f^n(z)) = \Phi_{-,f}(z) - \frac{1}{\alpha(f)} + n \]

(iv) when $f\in\mathcal{F}_1$ tends to $f_0$, $S_{\pm,f}\to S_{\pm,0}\cup \{0\}$ in the Hausdorff metric, and $\Phi_{\pm,f}\to \Phi_{\pm,0}$ in the topology defined in 2.5.1.

== Proposition 4.4.1 ==
[p. 356]

Let $f_0\in\mathcal{F}_0$, there exists a neighborhood $\mathcal{N}_0$ of $f_0$ (in the previously described "compact-open topology with domain of definition"), such that if $f\in \mathcal{N}_0 \cap \mathcal{F}_1$, then there exist large $\xi_0, \eta_0>0$, and analytic maps $\ph_f\colon D(\ph_f)\to\overline{\C}$ and $\tilde{\mathcal{R}}_f\colon D(\tilde{\mathcal{R}}_f)\to \C$ satisfying the following:
# $D(\ph_f)$ contains the set $$ \mathcal{Q}_f = \{ w\in\C\colon |\arg(-w-\xi_0)| < \frac{2\pi}{3}, |\arg(w+\frac{1}{\alpha(f)}-\xi_0)|<\frac{2\pi}{3} \} $$ and $\ph_f(D(\ph_f)) \subseteq D(f)$; If $w, w+1\in D(\ph_f)$ then $$ \ph_f(w+1)=f(\ph_f(w)) $$ $\ph_f(w)\to 0$ when $w\in \mathcal{Q}_f$ and $\Im(w)\to\infty$; $\ph_f(w)\to\sigma(f)$ when $w\in\mathcal{Q}_f$ and $\Im(w)\to -\infty$.
# $D(\tilde{\mathcal{R}}_f)$ contains $\{ w\in\C\colon |\Im(w)| > \eta_0\}$ and is invariant under $T(w)=w+1$; if $w\in D(\tilde{\mathcal{R}}_f)$ then $$\tilde{\mathcal{R}}_f(w+1)=\tilde{\mathcal{R}}(w)+1;$$ $\tilde{\mathcal{R}}_f(w)-w$ tends to $-\frac{1}{\alpha(f)}$ when $\Im(w)\to\infty$; to a constant when $\Im(w)\to -\infty$.
# If $w\in D(\tilde{\mathcal{R}}_f)\cap D(\ph_f)$ and $w'=\tilde{\mathcal{R}}_f(w)+n\in D(\ph_f)$ for some $n\in\Z$ then $$f^n(\ph_f(w))=\ph_f(w')\quad \text{for}\quad n\ge 0, \quad f^{-n}(\ph_f(w'))=\ph_f(w)\quad \text{for}\quad n<0$$ Moreover if $|\arg(w'+\frac{1}{2\alpha(f)}-\xi_0)|<\frac{2\pi}{3}$ then $n>0$.
# When $f\in\mathcal{F_1}$ and $f\to f_0$ $$\ph_f\to\ph_0\quad\text{and}\quad \tilde{\mathcal{R}}_f + \frac{1}{\alpha(f)} \to \tilde{\mathcal{E}}_{f_0}$$

== Proposition A.1 ==
In the settings of Proposition 3.2.2 (resp. 2.5.2), assume in addtion that $f_s(z)$ (resp. $F_s(z)$) is a family of holomorphic maps for $s\in \Delta \cup \{0\} \subset \mathbb{C}^k$, with $\Delta\subset\mathbb{C}^k$ an open connected set containing 0 in the boundary, such that $f_s$ (resp. $F_s(z)$) is continuous for $s\in\Delta\cup \{0\}$ and analytic in $s\in\Delta$, then the Fatou coordinates $\Phi_{\pm,f_s}$ (resp. $\Phi_{F_s}$) depend analytically on $s$ for $s\in\Delta$.

<references/>

Shishikura perturbed Fatou coordinates

2013-02-19T21:48:47Z

Bo198214: /* Proposition A.1 */ removed unnecessary notation

Shishikura writes<ref name="Shishikura2000">Shishikura, M. (2000). Bifurcation of parabolic fixed points. In Lei, Tan, The Mandelbrot set, theme and variations. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 274, 325-363 (2000)</ref> the following

== $\mathcal{F}_0$ ==
[p. 327]

If $f_0''(0)\neq 0$ by another coordinate change we may assume that $f_0''(0)=1$, so define
$$ \mathcal{F}_0 = \{ f_0 \colon f_0 \quad\text{is defined and analytic in a neighborhood of 0}\quad f_0(0)=0, f_0'(0)=1, f_0''(0)=1\}$$

== Neighborhood of $f$ in the compact-open topology with domain of definition ==
[ p. 332 ]

In what follows a map $f$ is always associated with its domain of definition $D(f)$. That is two maps are considered as distinct if they have distinct domains, even if the one is an extension of the other. A neighborhood of an analytic map $f\colon D(f)\to\overline{\C}$ is a set containing
$$\{ g \colon g \quad\text{is analytic},\quad D(g)\supset K, \sup_{z\in K} d(f(z)-g(z))<\eps \} $$
where $K$ is a compact set in $D(f)$, $\eps>0$ and $d(.,.)$ is the spherical metric.
A sequence $(f_n)$ converges to $f$ in this topology if and only if for any compact set $K\subset D(f)$ there exists $n_0$ such that $K\subset D(f_n)$ for all $n\ge n_0$ and $f_n|_K\to f|_K$ uniformly on $K$ as $n\to\infty$. The system of these neighborhoods define the "compact-open topology together with the domain of defintion", which is unfortunately not Hausdorff, since an extension of $f$ is contained in any neighborhood of $f$.

For $b_1,b_2\in\C$ with $\Re(b_1)<\Re(b_2)$ define
$$Q(b_1,b_2)=\{z\in\C\colon \Re(z-b_1)>-|\Im(z-b_1)|, \Re(z-b_2)>|\Im(z-b_2)|\}$$
If $b_1=-\infty$ (resp. $b_2=\infty$) then the corresponding condition should be removed.

== Proposition 2.5.2 ==
[p. 333]

Let $f$ be a holomorphic function defined on $Q=Q(b_1,b_2)$ with $\Re(b_2)>\Re(b_1)+2$ (here $b_1$ or $b_2$ may be $-\infty$ or $\infty$). Suppose
$$|F(z)-(z+1)|<\frac{1}{4},\quad |F'(z)-(z+1)|<\frac{1}{4}\quad \text{for all}\quad z\in Q.$$

Then
# $F$ is univalent [means injective] on $Q$.
# Let $z_0\in Q$ be a point such that $\Re(b_1)<\Re(z_0)<\Re(b_2)-\frac{5}{4}$. Denote by $S$ the closed region (a strip) bounded by the two curves $\ell=\{z_0+i y\colon y\in\R\}$ and $F(\ell)$. Then for any $z\in Q$ there exist a unique $n\in \Z$ such that $F^n(z)$ is defined and belongs to $S-F(\ell)$.
# There exists an univalent function $\Phi\colon Q\to \C$ satisfying $$\Phi(F(z))=\Phi(z)+1$$ whenever both sides are defined. Moreover $\Phi$ is unique up to an addition of a constant.
# Fix a point $z_0\in Q$. If we normalize $\Phi$ by $\Phi(z_0)=0$ then the correspondence $F\mapsto \Phi$ is continuous with respect to the compact-open topology.

== $\mathcal{F}$, $\mathcal{F}_1$ and $\alpha(f)$ ==
[p. 339]

$$ \mathcal{F} = \{ f \colon f \quad\text{is defined and analytic in a neighborhood of 0}\quad f(0)=0, f_0'(0)\neq 0\}$$
For $f\in \mathcal{F}$ we express the derivative
$$f'(0)=\exp(2\pi i \alpha(f))$$
where $\alpha(f)\in\C$ and $-\frac{1}{2} < \Re(\alpha(f)) \le \frac{1}{2}$ our study is focussed on the maps in the following class
$$ \mathcal{F}_1 = \{ f\in \mathcal{F}\colon |\arg(\alpha(f))| < \frac{\pi}{4}\} $$

=== Comments ===

By the definition we have $$
\begin{align*}
\log(f'(0))&=2\pi i\alpha(f)=2\pi i re^{i\beta}=2\pi r e^{i(\beta+\frac{\pi}{2})}\\
\arg(\log(f'(0))&=\beta+\frac{\pi}{2}
\end{align*}$$
so the condition for $\mathcal{F}_1$ is
$|\arg(\log(f'(0)))-\frac{\pi}{2}|<\frac{\pi}{4}$ or
$$\frac{\pi}{4} < \arg(\log(f'(0)) < \frac{3\pi}{4}$$

== $\sigma(f)$ ==
[p. 340]

Let $\sigma(f)$ be the other fixpoint of $f$ (besides 0).

...

$$\sigma(f) = 2\pi i \alpha(f)(1+o(1)), \quad\text{as}\quad f\to f_0$$

=== Comments ===

By the comment before, we can say that $\sigma(f)$ lies roughly in the sector of angle $(\frac{\pi}{4},\frac{3\pi}{4})$ above the tip 0.

== Proposition 3.2.2 ==
[p. 340]

Let $f_0\in\mathcal{F}_0$. There exists a neighborhood $\mathcal{N}_0$ of $f_0$ (in the sense of the topology defined in 2.5.1) such that if $f\in\mathcal{N}_0\cap \mathcal{F}_1$, then there exist closed Jordan domains $S_{-,f}$, $S_{+,f}$ and analytic functions $\Phi_{-,f}$, $\Phi_{+,f}$ satisfying the following:

(i) $S_{+,f}$ is bounded by an arc $\ell_{\pm}$ and its image $f(\ell_{\pm})$ such that $\ell_{\pm}$ joins two fixed points $\{0,\sigma(f)\}$, $\ell_{\pm}\cap f(\ell_{\pm}) = \{0,\sigma(f)\}$ and $S_{-,f}\cap S_{+,f} = \{0,\sigma(f)\}$;

(ii) $\Phi_{\pm,f}$ is defined, analytic and injective in a neighborhood of $S'_{\pm,f} = S_{\pm,f}\setminus \{0,\sigma(f)\}$;

(iii) if $z\in S'_{-,f}$ then there is an $n\ge 1$ such that $f^n(z)\in S'_{+,f}$ and for the smallest such $n$,
\[ \Phi_{+,f}(f^n(z)) = \Phi_{-,f}(z) - \frac{1}{\alpha(f)} + n \]

(iv) when $f\in\mathcal{F}_1$ tends to $f_0$, $S_{\pm,f}\to S_{\pm,0}\cup \{0\}$ in the Hausdorff metric, and $\Phi_{\pm,f}\to \Phi_{\pm,0}$ in the topology defined in 2.5.1.

== Proposition 4.4.1 ==
[p. 356]

Let $f_0\in\mathcal{F}_0$, there exists a neighborhood $\mathcal{N}_0$ of $f_0$ (in the previously described "compact-open topology with domain of definition"), such that if $f\in \mathcal{N}_0 \cap \mathcal{F}_1$, then there exist large $\xi_0, \eta_0>0$, and analytic maps $\ph_f\colon D(\ph_f)\to\overline{\C}$ and $\tilde{\mathcal{R}}_f\colon D(\tilde{\mathcal{R}}_f)\to \C$ satisfying the following:
# $D(\ph_f)$ contains the set $$ \mathcal{Q}_f = \{ w\in\C\colon |\arg(-w-\xi_0)| < \frac{2\pi}{3}, |\arg(w+\frac{1}{\alpha(f)}-\xi_0)|<\frac{2\pi}{3} \} $$ and $\ph_f(D(\ph_f)) \subseteq D(f)$; If $w, w+1\in D(\ph_f)$ then $$ \ph_f(w+1)=f(\ph_f(w)) $$ $\ph_f(w)\to 0$ when $w\in \mathcal{Q}_f$ and $\Im(w)\to\infty$; $\ph_f(w)\to\sigma(f)$ when $w\in\mathcal{Q}_f$ and $\Im(w)\to -\infty$.
# $D(\tilde{\mathcal{R}}_f)$ contains $\{ w\in\C\colon |\Im(w)| > \eta_0\}$ and is invariant under $T(w)=w+1$; if $w\in D(\tilde{\mathcal{R}}_f)$ then $$\tilde{\mathcal{R}}_f(w+1)=\tilde{\mathcal{R}}(w)+1;$$ $\tilde{\mathcal{R}}_f(w)-w$ tends to $-\frac{1}{\alpha(f)}$ when $\Im(w)\to\infty$; to a constant when $\Im(w)\to -\infty$.
# If $w\in D(\tilde{\mathcal{R}}_f)\cap D(\ph_f)$ and $w'=\tilde{\mathcal{R}}_f(w)+n\in D(\ph_f)$ for some $n\in\Z$ then $$f^n(\ph_f(w))=\ph_f(w')\quad \text{for}\quad n\ge 0, \quad f^{-n}(\ph_f(w'))=\ph_f(w)\quad \text{for}\quad n<0$$ Moreover if $|\arg(w'+\frac{1}{2\alpha(f)}-\xi_0)|<\frac{2\pi}{3}$ then $n>0$.
# When $f\in\mathcal{F_1}$ and $f\to f_0$ $$\ph_f\to\ph_0\quad\text{and}\quad \tilde{\mathcal{R}}_f + \frac{1}{\alpha(f)} \to \tilde{\mathcal{E}}_{f_0}$$

== Proposition A.1 ==
In the settings of Proposition 3.2.2 (resp. 2.5.2), assume in addtion that $f_s(z)$ (resp. $F_s(z)$) is a family of holomorphic maps for $s\in \Delta \cup \{0\} \subset \mathbb{C}^k$, with $\Delta\subset\mathbb{C}^k$ an open connected set containing 0 in the boundary, such that $f_s$ (resp. $F_s(z)$) is continuous for $s\in\Delta\cup \{0\}$ and analytic in $s\in\Delta$, the the Fatou coordinates $\Phi_{\pm,f_s}$ (resp. $\Phi_{F_s}$) depend analytically on $s$ for $s\in\Delta$.

<references/>

Shishikura perturbed Fatou coordinates

2013-02-19T21:46:58Z

Bo198214: /* Proposition A.1 */ wrote Proposition A.1

Shishikura writes<ref name="Shishikura2000">Shishikura, M. (2000). Bifurcation of parabolic fixed points. In Lei, Tan, The Mandelbrot set, theme and variations. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 274, 325-363 (2000)</ref> the following

== $\mathcal{F}_0$ ==
[p. 327]

If $f_0''(0)\neq 0$ by another coordinate change we may assume that $f_0''(0)=1$, so define
$$ \mathcal{F}_0 = \{ f_0 \colon f_0 \quad\text{is defined and analytic in a neighborhood of 0}\quad f_0(0)=0, f_0'(0)=1, f_0''(0)=1\}$$

== Neighborhood of $f$ in the compact-open topology with domain of definition ==
[ p. 332 ]

In what follows a map $f$ is always associated with its domain of definition $D(f)$. That is two maps are considered as distinct if they have distinct domains, even if the one is an extension of the other. A neighborhood of an analytic map $f\colon D(f)\to\overline{\C}$ is a set containing
$$\{ g \colon g \quad\text{is analytic},\quad D(g)\supset K, \sup_{z\in K} d(f(z)-g(z))<\eps \} $$
where $K$ is a compact set in $D(f)$, $\eps>0$ and $d(.,.)$ is the spherical metric.
A sequence $(f_n)$ converges to $f$ in this topology if and only if for any compact set $K\subset D(f)$ there exists $n_0$ such that $K\subset D(f_n)$ for all $n\ge n_0$ and $f_n|_K\to f|_K$ uniformly on $K$ as $n\to\infty$. The system of these neighborhoods define the "compact-open topology together with the domain of defintion", which is unfortunately not Hausdorff, since an extension of $f$ is contained in any neighborhood of $f$.

For $b_1,b_2\in\C$ with $\Re(b_1)<\Re(b_2)$ define
$$Q(b_1,b_2)=\{z\in\C\colon \Re(z-b_1)>-|\Im(z-b_1)|, \Re(z-b_2)>|\Im(z-b_2)|\}$$
If $b_1=-\infty$ (resp. $b_2=\infty$) then the corresponding condition should be removed.

== Proposition 2.5.2 ==
[p. 333]

Let $f$ be a holomorphic function defined on $Q=Q(b_1,b_2)$ with $\Re(b_2)>\Re(b_1)+2$ (here $b_1$ or $b_2$ may be $-\infty$ or $\infty$). Suppose
$$|F(z)-(z+1)|<\frac{1}{4},\quad |F'(z)-(z+1)|<\frac{1}{4}\quad \text{for all}\quad z\in Q.$$

Then
# $F$ is univalent [means injective] on $Q$.
# Let $z_0\in Q$ be a point such that $\Re(b_1)<\Re(z_0)<\Re(b_2)-\frac{5}{4}$. Denote by $S$ the closed region (a strip) bounded by the two curves $\ell=\{z_0+i y\colon y\in\R\}$ and $F(\ell)$. Then for any $z\in Q$ there exist a unique $n\in \Z$ such that $F^n(z)$ is defined and belongs to $S-F(\ell)$.
# There exists an univalent function $\Phi\colon Q\to \C$ satisfying $$\Phi(F(z))=\Phi(z)+1$$ whenever both sides are defined. Moreover $\Phi$ is unique up to an addition of a constant.
# Fix a point $z_0\in Q$. If we normalize $\Phi$ by $\Phi(z_0)=0$ then the correspondence $F\mapsto \Phi$ is continuous with respect to the compact-open topology.

== $\mathcal{F}$, $\mathcal{F}_1$ and $\alpha(f)$ ==
[p. 339]

$$ \mathcal{F} = \{ f \colon f \quad\text{is defined and analytic in a neighborhood of 0}\quad f(0)=0, f_0'(0)\neq 0\}$$
For $f\in \mathcal{F}$ we express the derivative
$$f'(0)=\exp(2\pi i \alpha(f))$$
where $\alpha(f)\in\C$ and $-\frac{1}{2} < \Re(\alpha(f)) \le \frac{1}{2}$ our study is focussed on the maps in the following class
$$ \mathcal{F}_1 = \{ f\in \mathcal{F}\colon |\arg(\alpha(f))| < \frac{\pi}{4}\} $$

=== Comments ===

By the definition we have $$
\begin{align*}
\log(f'(0))&=2\pi i\alpha(f)=2\pi i re^{i\beta}=2\pi r e^{i(\beta+\frac{\pi}{2})}\\
\arg(\log(f'(0))&=\beta+\frac{\pi}{2}
\end{align*}$$
so the condition for $\mathcal{F}_1$ is
$|\arg(\log(f'(0)))-\frac{\pi}{2}|<\frac{\pi}{4}$ or
$$\frac{\pi}{4} < \arg(\log(f'(0)) < \frac{3\pi}{4}$$

== $\sigma(f)$ ==
[p. 340]

Let $\sigma(f)$ be the other fixpoint of $f$ (besides 0).

...

$$\sigma(f) = 2\pi i \alpha(f)(1+o(1)), \quad\text{as}\quad f\to f_0$$

=== Comments ===

By the comment before, we can say that $\sigma(f)$ lies roughly in the sector of angle $(\frac{\pi}{4},\frac{3\pi}{4})$ above the tip 0.

== Proposition 3.2.2 ==
[p. 340]

Let $f_0\in\mathcal{F}_0$. There exists a neighborhood $\mathcal{N}_0$ of $f_0$ (in the sense of the topology defined in 2.5.1) such that if $f\in\mathcal{N}_0\cap \mathcal{F}_1$, then there exist closed Jordan domains $S_{-,f}$, $S_{+,f}$ and analytic functions $\Phi_{-,f}$, $\Phi_{+,f}$ satisfying the following:

(i) $S_{+,f}$ is bounded by an arc $\ell_{\pm}$ and its image $f(\ell_{\pm})$ such that $\ell_{\pm}$ joins two fixed points $\{0,\sigma(f)\}$, $\ell_{\pm}\cap f(\ell_{\pm}) = \{0,\sigma(f)\}$ and $S_{-,f}\cap S_{+,f} = \{0,\sigma(f)\}$;

(ii) $\Phi_{\pm,f}$ is defined, analytic and injective in a neighborhood of $S'_{\pm,f} = S_{\pm,f}\setminus \{0,\sigma(f)\}$;

(iii) if $z\in S'_{-,f}$ then there is an $n\ge 1$ such that $f^n(z)\in S'_{+,f}$ and for the smallest such $n$,
\[ \Phi_{+,f}(f^n(z)) = \Phi_{-,f}(z) - \frac{1}{\alpha(f)} + n \]

(iv) when $f\in\mathcal{F}_1$ tends to $f_0$, $S_{\pm,f}\to S_{\pm,0}\cup \{0\}$ in the Hausdorff metric, and $\Phi_{\pm,f}\to \Phi_{\pm,0}$ in the topology defined in 2.5.1.

== Proposition 4.4.1 ==
[p. 356]

Let $f_0\in\mathcal{F}_0$, there exists a neighborhood $\mathcal{N}_0$ of $f_0$ (in the previously described "compact-open topology with domain of definition"), such that if $f\in \mathcal{N}_0 \cap \mathcal{F}_1$, then there exist large $\xi_0, \eta_0>0$, and analytic maps $\ph_f\colon D(\ph_f)\to\overline{\C}$ and $\tilde{\mathcal{R}}_f\colon D(\tilde{\mathcal{R}}_f)\to \C$ satisfying the following:
# $D(\ph_f)$ contains the set $$ \mathcal{Q}_f = \{ w\in\C\colon |\arg(-w-\xi_0)| < \frac{2\pi}{3}, |\arg(w+\frac{1}{\alpha(f)}-\xi_0)|<\frac{2\pi}{3} \} $$ and $\ph_f(D(\ph_f)) \subseteq D(f)$; If $w, w+1\in D(\ph_f)$ then $$ \ph_f(w+1)=f(\ph_f(w)) $$ $\ph_f(w)\to 0$ when $w\in \mathcal{Q}_f$ and $\Im(w)\to\infty$; $\ph_f(w)\to\sigma(f)$ when $w\in\mathcal{Q}_f$ and $\Im(w)\to -\infty$.
# $D(\tilde{\mathcal{R}}_f)$ contains $\{ w\in\C\colon |\Im(w)| > \eta_0\}$ and is invariant under $T(w)=w+1$; if $w\in D(\tilde{\mathcal{R}}_f)$ then $$\tilde{\mathcal{R}}_f(w+1)=\tilde{\mathcal{R}}(w)+1;$$ $\tilde{\mathcal{R}}_f(w)-w$ tends to $-\frac{1}{\alpha(f)}$ when $\Im(w)\to\infty$; to a constant when $\Im(w)\to -\infty$.
# If $w\in D(\tilde{\mathcal{R}}_f)\cap D(\ph_f)$ and $w'=\tilde{\mathcal{R}}_f(w)+n\in D(\ph_f)$ for some $n\in\Z$ then $$f^n(\ph_f(w))=\ph_f(w')\quad \text{for}\quad n\ge 0, \quad f^{-n}(\ph_f(w'))=\ph_f(w)\quad \text{for}\quad n<0$$ Moreover if $|\arg(w'+\frac{1}{2\alpha(f)}-\xi_0)|<\frac{2\pi}{3}$ then $n>0$.
# When $f\in\mathcal{F_1}$ and $f\to f_0$ $$\ph_f\to\ph_0\quad\text{and}\quad \tilde{\mathcal{R}}_f + \frac{1}{\alpha(f)} \to \tilde{\mathcal{E}}_{f_0}$$

== Proposition A.1 ==
In the settings of Proposition 3.2.2 (resp. 2.5.2), assume in addtion that $f_s(z)$ (resp. $F_s(z)$) is a family of holomorphic maps for $s\in \Delta \cup \{0\} \subset \mathbb{C}^k$, with $\Delta\subset\mathbb{C}^k$ an open connected set containing 0 in the boundary, such that $f_s$ (resp. $F_s(z)$) is continuous for $s\in\Delta\cup \{0\}$ and analytic in $s\in\Delta$, the the Fatou coordinates $\Phi_{\pm,f_s} := \Phi_{\pm,s}$ (resp. $\Phi_s := \Phi_{F_s}$) depend analytically on $s$ for $s\in\Delta$.

<references/>

Shishikura perturbed Fatou coordinates

2013-02-19T21:40:18Z

Bo198214: added proposition A.1

Shishikura writes<ref name="Shishikura2000">Shishikura, M. (2000). Bifurcation of parabolic fixed points. In Lei, Tan, The Mandelbrot set, theme and variations. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 274, 325-363 (2000)</ref> the following

== $\mathcal{F}_0$ ==
[p. 327]

If $f_0''(0)\neq 0$ by another coordinate change we may assume that $f_0''(0)=1$, so define
$$ \mathcal{F}_0 = \{ f_0 \colon f_0 \quad\text{is defined and analytic in a neighborhood of 0}\quad f_0(0)=0, f_0'(0)=1, f_0''(0)=1\}$$

== Neighborhood of $f$ in the compact-open topology with domain of definition ==
[ p. 332 ]

In what follows a map $f$ is always associated with its domain of definition $D(f)$. That is two maps are considered as distinct if they have distinct domains, even if the one is an extension of the other. A neighborhood of an analytic map $f\colon D(f)\to\overline{\C}$ is a set containing
$$\{ g \colon g \quad\text{is analytic},\quad D(g)\supset K, \sup_{z\in K} d(f(z)-g(z))<\eps \} $$
where $K$ is a compact set in $D(f)$, $\eps>0$ and $d(.,.)$ is the spherical metric.
A sequence $(f_n)$ converges to $f$ in this topology if and only if for any compact set $K\subset D(f)$ there exists $n_0$ such that $K\subset D(f_n)$ for all $n\ge n_0$ and $f_n|_K\to f|_K$ uniformly on $K$ as $n\to\infty$. The system of these neighborhoods define the "compact-open topology together with the domain of defintion", which is unfortunately not Hausdorff, since an extension of $f$ is contained in any neighborhood of $f$.

For $b_1,b_2\in\C$ with $\Re(b_1)<\Re(b_2)$ define
$$Q(b_1,b_2)=\{z\in\C\colon \Re(z-b_1)>-|\Im(z-b_1)|, \Re(z-b_2)>|\Im(z-b_2)|\}$$
If $b_1=-\infty$ (resp. $b_2=\infty$) then the corresponding condition should be removed.

== Proposition 2.5.2 ==
[p. 333]

Let $f$ be a holomorphic function defined on $Q=Q(b_1,b_2)$ with $\Re(b_2)>\Re(b_1)+2$ (here $b_1$ or $b_2$ may be $-\infty$ or $\infty$). Suppose
$$|F(z)-(z+1)|<\frac{1}{4},\quad |F'(z)-(z+1)|<\frac{1}{4}\quad \text{for all}\quad z\in Q.$$

Then
# $F$ is univalent [means injective] on $Q$.
# Let $z_0\in Q$ be a point such that $\Re(b_1)<\Re(z_0)<\Re(b_2)-\frac{5}{4}$. Denote by $S$ the closed region (a strip) bounded by the two curves $\ell=\{z_0+i y\colon y\in\R\}$ and $F(\ell)$. Then for any $z\in Q$ there exist a unique $n\in \Z$ such that $F^n(z)$ is defined and belongs to $S-F(\ell)$.
# There exists an univalent function $\Phi\colon Q\to \C$ satisfying $$\Phi(F(z))=\Phi(z)+1$$ whenever both sides are defined. Moreover $\Phi$ is unique up to an addition of a constant.
# Fix a point $z_0\in Q$. If we normalize $\Phi$ by $\Phi(z_0)=0$ then the correspondence $F\mapsto \Phi$ is continuous with respect to the compact-open topology.

== $\mathcal{F}$, $\mathcal{F}_1$ and $\alpha(f)$ ==
[p. 339]

$$ \mathcal{F} = \{ f \colon f \quad\text{is defined and analytic in a neighborhood of 0}\quad f(0)=0, f_0'(0)\neq 0\}$$
For $f\in \mathcal{F}$ we express the derivative
$$f'(0)=\exp(2\pi i \alpha(f))$$
where $\alpha(f)\in\C$ and $-\frac{1}{2} < \Re(\alpha(f)) \le \frac{1}{2}$ our study is focussed on the maps in the following class
$$ \mathcal{F}_1 = \{ f\in \mathcal{F}\colon |\arg(\alpha(f))| < \frac{\pi}{4}\} $$

=== Comments ===

By the definition we have $$
\begin{align*}
\log(f'(0))&=2\pi i\alpha(f)=2\pi i re^{i\beta}=2\pi r e^{i(\beta+\frac{\pi}{2})}\\
\arg(\log(f'(0))&=\beta+\frac{\pi}{2}
\end{align*}$$
so the condition for $\mathcal{F}_1$ is
$|\arg(\log(f'(0)))-\frac{\pi}{2}|<\frac{\pi}{4}$ or
$$\frac{\pi}{4} < \arg(\log(f'(0)) < \frac{3\pi}{4}$$

== $\sigma(f)$ ==
[p. 340]

Let $\sigma(f)$ be the other fixpoint of $f$ (besides 0).

...

$$\sigma(f) = 2\pi i \alpha(f)(1+o(1)), \quad\text{as}\quad f\to f_0$$

=== Comments ===

By the comment before, we can say that $\sigma(f)$ lies roughly in the sector of angle $(\frac{\pi}{4},\frac{3\pi}{4})$ above the tip 0.

== Proposition 3.2.2 ==
[p. 340]

Let $f_0\in\mathcal{F}_0$. There exists a neighborhood $\mathcal{N}_0$ of $f_0$ (in the sense of the topology defined in 2.5.1) such that if $f\in\mathcal{N}_0\cap \mathcal{F}_1$, then there exist closed Jordan domains $S_{-,f}$, $S_{+,f}$ and analytic functions $\Phi_{-,f}$, $\Phi_{+,f}$ satisfying the following:

(i) $S_{+,f}$ is bounded by an arc $\ell_{\pm}$ and its image $f(\ell_{\pm})$ such that $\ell_{\pm}$ joins two fixed points $\{0,\sigma(f)\}$, $\ell_{\pm}\cap f(\ell_{\pm}) = \{0,\sigma(f)\}$ and $S_{-,f}\cap S_{+,f} = \{0,\sigma(f)\}$;

(ii) $\Phi_{\pm,f}$ is defined, analytic and injective in a neighborhood of $S'_{\pm,f} = S_{\pm,f}\setminus \{0,\sigma(f)\}$;

(iii) if $z\in S'_{-,f}$ then there is an $n\ge 1$ such that $f^n(z)\in S'_{+,f}$ and for the smallest such $n$,
\[ \Phi_{+,f}(f^n(z)) = \Phi_{-,f}(z) - \frac{1}{\alpha(f)} + n \]

(iv) when $f\in\mathcal{F}_1$ tends to $f_0$, $S_{\pm,f}\to S_{\pm,0}\cup \{0\}$ in the Hausdorff metric, and $\Phi_{\pm,f}\to \Phi_{\pm,0}$ in the topology defined in 2.5.1.

== Proposition 4.4.1 ==
[p. 356]

Let $f_0\in\mathcal{F}_0$, there exists a neighborhood $\mathcal{N}_0$ of $f_0$ (in the previously described "compact-open topology with domain of definition"), such that if $f\in \mathcal{N}_0 \cap \mathcal{F}_1$, then there exist large $\xi_0, \eta_0>0$, and analytic maps $\ph_f\colon D(\ph_f)\to\overline{\C}$ and $\tilde{\mathcal{R}}_f\colon D(\tilde{\mathcal{R}}_f)\to \C$ satisfying the following:
# $D(\ph_f)$ contains the set $$ \mathcal{Q}_f = \{ w\in\C\colon |\arg(-w-\xi_0)| < \frac{2\pi}{3}, |\arg(w+\frac{1}{\alpha(f)}-\xi_0)|<\frac{2\pi}{3} \} $$ and $\ph_f(D(\ph_f)) \subseteq D(f)$; If $w, w+1\in D(\ph_f)$ then $$ \ph_f(w+1)=f(\ph_f(w)) $$ $\ph_f(w)\to 0$ when $w\in \mathcal{Q}_f$ and $\Im(w)\to\infty$; $\ph_f(w)\to\sigma(f)$ when $w\in\mathcal{Q}_f$ and $\Im(w)\to -\infty$.
# $D(\tilde{\mathcal{R}}_f)$ contains $\{ w\in\C\colon |\Im(w)| > \eta_0\}$ and is invariant under $T(w)=w+1$; if $w\in D(\tilde{\mathcal{R}}_f)$ then $$\tilde{\mathcal{R}}_f(w+1)=\tilde{\mathcal{R}}(w)+1;$$ $\tilde{\mathcal{R}}_f(w)-w$ tends to $-\frac{1}{\alpha(f)}$ when $\Im(w)\to\infty$; to a constant when $\Im(w)\to -\infty$.
# If $w\in D(\tilde{\mathcal{R}}_f)\cap D(\ph_f)$ and $w'=\tilde{\mathcal{R}}_f(w)+n\in D(\ph_f)$ for some $n\in\Z$ then $$f^n(\ph_f(w))=\ph_f(w')\quad \text{for}\quad n\ge 0, \quad f^{-n}(\ph_f(w'))=\ph_f(w)\quad \text{for}\quad n<0$$ Moreover if $|\arg(w'+\frac{1}{2\alpha(f)}-\xi_0)|<\frac{2\pi}{3}$ then $n>0$.
# When $f\in\mathcal{F_1}$ and $f\to f_0$ $$\ph_f\to\ph_0\quad\text{and}\quad \tilde{\mathcal{R}}_f + \frac{1}{\alpha(f)} \to \tilde{\mathcal{E}}_{f_0}$$

== Proposition A.1 ==

<references/>

Shishikura perturbed Fatou coordinates

2013-02-19T21:39:06Z

Bo198214: /* Proposition 3.2.2 */

Shishikura writes<ref name="Shishikura2000">Shishikura, M. (2000). Bifurcation of parabolic fixed points. In Lei, Tan, The Mandelbrot set, theme and variations. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 274, 325-363 (2000)</ref> the following

== $\mathcal{F}_0$ ==
[p. 327]

If $f_0''(0)\neq 0$ by another coordinate change we may assume that $f_0''(0)=1$, so define
$$ \mathcal{F}_0 = \{ f_0 \colon f_0 \quad\text{is defined and analytic in a neighborhood of 0}\quad f_0(0)=0, f_0'(0)=1, f_0''(0)=1\}$$

== Neighborhood of $f$ in the compact-open topology with domain of definition ==
[ p. 332 ]

In what follows a map $f$ is always associated with its domain of definition $D(f)$. That is two maps are considered as distinct if they have distinct domains, even if the one is an extension of the other. A neighborhood of an analytic map $f\colon D(f)\to\overline{\C}$ is a set containing
$$\{ g \colon g \quad\text{is analytic},\quad D(g)\supset K, \sup_{z\in K} d(f(z)-g(z))<\eps \} $$
where $K$ is a compact set in $D(f)$, $\eps>0$ and $d(.,.)$ is the spherical metric.
A sequence $(f_n)$ converges to $f$ in this topology if and only if for any compact set $K\subset D(f)$ there exists $n_0$ such that $K\subset D(f_n)$ for all $n\ge n_0$ and $f_n|_K\to f|_K$ uniformly on $K$ as $n\to\infty$. The system of these neighborhoods define the "compact-open topology together with the domain of defintion", which is unfortunately not Hausdorff, since an extension of $f$ is contained in any neighborhood of $f$.

For $b_1,b_2\in\C$ with $\Re(b_1)<\Re(b_2)$ define
$$Q(b_1,b_2)=\{z\in\C\colon \Re(z-b_1)>-|\Im(z-b_1)|, \Re(z-b_2)>|\Im(z-b_2)|\}$$
If $b_1=-\infty$ (resp. $b_2=\infty$) then the corresponding condition should be removed.

== Proposition 2.5.2 ==
[p. 333]

Let $f$ be a holomorphic function defined on $Q=Q(b_1,b_2)$ with $\Re(b_2)>\Re(b_1)+2$ (here $b_1$ or $b_2$ may be $-\infty$ or $\infty$). Suppose
$$|F(z)-(z+1)|<\frac{1}{4},\quad |F'(z)-(z+1)|<\frac{1}{4}\quad \text{for all}\quad z\in Q.$$

Then
# $F$ is univalent [means injective] on $Q$.
# Let $z_0\in Q$ be a point such that $\Re(b_1)<\Re(z_0)<\Re(b_2)-\frac{5}{4}$. Denote by $S$ the closed region (a strip) bounded by the two curves $\ell=\{z_0+i y\colon y\in\R\}$ and $F(\ell)$. Then for any $z\in Q$ there exist a unique $n\in \Z$ such that $F^n(z)$ is defined and belongs to $S-F(\ell)$.
# There exists an univalent function $\Phi\colon Q\to \C$ satisfying $$\Phi(F(z))=\Phi(z)+1$$ whenever both sides are defined. Moreover $\Phi$ is unique up to an addition of a constant.
# Fix a point $z_0\in Q$. If we normalize $\Phi$ by $\Phi(z_0)=0$ then the correspondence $F\mapsto \Phi$ is continuous with respect to the compact-open topology.

== $\mathcal{F}$, $\mathcal{F}_1$ and $\alpha(f)$ ==
[p. 339]

$$ \mathcal{F} = \{ f \colon f \quad\text{is defined and analytic in a neighborhood of 0}\quad f(0)=0, f_0'(0)\neq 0\}$$
For $f\in \mathcal{F}$ we express the derivative
$$f'(0)=\exp(2\pi i \alpha(f))$$
where $\alpha(f)\in\C$ and $-\frac{1}{2} < \Re(\alpha(f)) \le \frac{1}{2}$ our study is focussed on the maps in the following class
$$ \mathcal{F}_1 = \{ f\in \mathcal{F}\colon |\arg(\alpha(f))| < \frac{\pi}{4}\} $$

=== Comments ===

By the definition we have $$
\begin{align*}
\log(f'(0))&=2\pi i\alpha(f)=2\pi i re^{i\beta}=2\pi r e^{i(\beta+\frac{\pi}{2})}\\
\arg(\log(f'(0))&=\beta+\frac{\pi}{2}
\end{align*}$$
so the condition for $\mathcal{F}_1$ is
$|\arg(\log(f'(0)))-\frac{\pi}{2}|<\frac{\pi}{4}$ or
$$\frac{\pi}{4} < \arg(\log(f'(0)) < \frac{3\pi}{4}$$

== $\sigma(f)$ ==
[p. 340]

Let $\sigma(f)$ be the other fixpoint of $f$ (besides 0).

...

$$\sigma(f) = 2\pi i \alpha(f)(1+o(1)), \quad\text{as}\quad f\to f_0$$

=== Comments ===

By the comment before, we can say that $\sigma(f)$ lies roughly in the sector of angle $(\frac{\pi}{4},\frac{3\pi}{4})$ above the tip 0.

== Proposition 3.2.2 ==
[p. 340]

Let $f_0\in\mathcal{F}_0$. There exists a neighborhood $\mathcal{N}_0$ of $f_0$ (in the sense of the topology defined in 2.5.1) such that if $f\in\mathcal{N}_0\cap \mathcal{F}_1$, then there exist closed Jordan domains $S_{-,f}$, $S_{+,f}$ and analytic functions $\Phi_{-,f}$, $\Phi_{+,f}$ satisfying the following:

(i) $S_{+,f}$ is bounded by an arc $\ell_{\pm}$ and its image $f(\ell_{\pm})$ such that $\ell_{\pm}$ joins two fixed points $\{0,\sigma(f)\}$, $\ell_{\pm}\cap f(\ell_{\pm}) = \{0,\sigma(f)\}$ and $S_{-,f}\cap S_{+,f} = \{0,\sigma(f)\}$;

(ii) $\Phi_{\pm,f}$ is defined, analytic and injective in a neighborhood of $S'_{\pm,f} = S_{\pm,f}\setminus \{0,\sigma(f)\}$;

(iii) if $z\in S'_{-,f}$ then there is an $n\ge 1$ such that $f^n(z)\in S'_{+,f}$ and for the smallest such $n$,
\[ \Phi_{+,f}(f^n(z)) = \Phi_{-,f}(z) - \frac{1}{\alpha(f)} + n \]

(iv) when $f\in\mathcal{F}_1$ tends to $f_0$, $S_{\pm,f}\to S_{\pm,0}\cup \{0\}$ in the Hausdorff metric, and $\Phi_{\pm,f}\to \Phi_{\pm,0}$ in the topology defined in 2.5.1.

== Proposition 4.4.1 ==
[p. 356]

Let $f_0\in\mathcal{F}_0$, there exists a neighborhood $\mathcal{N}_0$ of $f_0$ (in the previously described "compact-open topology with domain of definition"), such that if $f\in \mathcal{N}_0 \cap \mathcal{F}_1$, then there exist large $\xi_0, \eta_0>0$, and analytic maps $\ph_f\colon D(\ph_f)\to\overline{\C}$ and $\tilde{\mathcal{R}}_f\colon D(\tilde{\mathcal{R}}_f)\to \C$ satisfying the following:
# $D(\ph_f)$ contains the set $$ \mathcal{Q}_f = \{ w\in\C\colon |\arg(-w-\xi_0)| < \frac{2\pi}{3}, |\arg(w+\frac{1}{\alpha(f)}-\xi_0)|<\frac{2\pi}{3} \} $$ and $\ph_f(D(\ph_f)) \subseteq D(f)$; If $w, w+1\in D(\ph_f)$ then $$ \ph_f(w+1)=f(\ph_f(w)) $$ $\ph_f(w)\to 0$ when $w\in \mathcal{Q}_f$ and $\Im(w)\to\infty$; $\ph_f(w)\to\sigma(f)$ when $w\in\mathcal{Q}_f$ and $\Im(w)\to -\infty$.
# $D(\tilde{\mathcal{R}}_f)$ contains $\{ w\in\C\colon |\Im(w)| > \eta_0\}$ and is invariant under $T(w)=w+1$; if $w\in D(\tilde{\mathcal{R}}_f)$ then $$\tilde{\mathcal{R}}_f(w+1)=\tilde{\mathcal{R}}(w)+1;$$ $\tilde{\mathcal{R}}_f(w)-w$ tends to $-\frac{1}{\alpha(f)}$ when $\Im(w)\to\infty$; to a constant when $\Im(w)\to -\infty$.
# If $w\in D(\tilde{\mathcal{R}}_f)\cap D(\ph_f)$ and $w'=\tilde{\mathcal{R}}_f(w)+n\in D(\ph_f)$ for some $n\in\Z$ then $$f^n(\ph_f(w))=\ph_f(w')\quad \text{for}\quad n\ge 0, \quad f^{-n}(\ph_f(w'))=\ph_f(w)\quad \text{for}\quad n<0$$ Moreover if $|\arg(w'+\frac{1}{2\alpha(f)}-\xi_0)|<\frac{2\pi}{3}$ then $n>0$.
# When $f\in\mathcal{F_1}$ and $f\to f_0$ $$\ph_f\to\ph_0\quad\text{and}\quad \tilde{\mathcal{R}}_f + \frac{1}{\alpha(f)} \to \tilde{\mathcal{E}}_{f_0}$$

<references/>

Shishikura perturbed Fatou coordinates

2013-02-19T21:38:40Z

Bo198214: /* Proposition 3.2.2 */ added page number

Shishikura writes<ref name="Shishikura2000">Shishikura, M. (2000). Bifurcation of parabolic fixed points. In Lei, Tan, The Mandelbrot set, theme and variations. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 274, 325-363 (2000)</ref> the following

== $\mathcal{F}_0$ ==
[p. 327]

If $f_0''(0)\neq 0$ by another coordinate change we may assume that $f_0''(0)=1$, so define
$$ \mathcal{F}_0 = \{ f_0 \colon f_0 \quad\text{is defined and analytic in a neighborhood of 0}\quad f_0(0)=0, f_0'(0)=1, f_0''(0)=1\}$$

== Neighborhood of $f$ in the compact-open topology with domain of definition ==
[ p. 332 ]

In what follows a map $f$ is always associated with its domain of definition $D(f)$. That is two maps are considered as distinct if they have distinct domains, even if the one is an extension of the other. A neighborhood of an analytic map $f\colon D(f)\to\overline{\C}$ is a set containing
$$\{ g \colon g \quad\text{is analytic},\quad D(g)\supset K, \sup_{z\in K} d(f(z)-g(z))<\eps \} $$
where $K$ is a compact set in $D(f)$, $\eps>0$ and $d(.,.)$ is the spherical metric.
A sequence $(f_n)$ converges to $f$ in this topology if and only if for any compact set $K\subset D(f)$ there exists $n_0$ such that $K\subset D(f_n)$ for all $n\ge n_0$ and $f_n|_K\to f|_K$ uniformly on $K$ as $n\to\infty$. The system of these neighborhoods define the "compact-open topology together with the domain of defintion", which is unfortunately not Hausdorff, since an extension of $f$ is contained in any neighborhood of $f$.

For $b_1,b_2\in\C$ with $\Re(b_1)<\Re(b_2)$ define
$$Q(b_1,b_2)=\{z\in\C\colon \Re(z-b_1)>-|\Im(z-b_1)|, \Re(z-b_2)>|\Im(z-b_2)|\}$$
If $b_1=-\infty$ (resp. $b_2=\infty$) then the corresponding condition should be removed.

== Proposition 2.5.2 ==
[p. 333]

Let $f$ be a holomorphic function defined on $Q=Q(b_1,b_2)$ with $\Re(b_2)>\Re(b_1)+2$ (here $b_1$ or $b_2$ may be $-\infty$ or $\infty$). Suppose
$$|F(z)-(z+1)|<\frac{1}{4},\quad |F'(z)-(z+1)|<\frac{1}{4}\quad \text{for all}\quad z\in Q.$$

Then
# $F$ is univalent [means injective] on $Q$.
# Let $z_0\in Q$ be a point such that $\Re(b_1)<\Re(z_0)<\Re(b_2)-\frac{5}{4}$. Denote by $S$ the closed region (a strip) bounded by the two curves $\ell=\{z_0+i y\colon y\in\R\}$ and $F(\ell)$. Then for any $z\in Q$ there exist a unique $n\in \Z$ such that $F^n(z)$ is defined and belongs to $S-F(\ell)$.
# There exists an univalent function $\Phi\colon Q\to \C$ satisfying $$\Phi(F(z))=\Phi(z)+1$$ whenever both sides are defined. Moreover $\Phi$ is unique up to an addition of a constant.
# Fix a point $z_0\in Q$. If we normalize $\Phi$ by $\Phi(z_0)=0$ then the correspondence $F\mapsto \Phi$ is continuous with respect to the compact-open topology.

== $\mathcal{F}$, $\mathcal{F}_1$ and $\alpha(f)$ ==
[p. 339]

$$ \mathcal{F} = \{ f \colon f \quad\text{is defined and analytic in a neighborhood of 0}\quad f(0)=0, f_0'(0)\neq 0\}$$
For $f\in \mathcal{F}$ we express the derivative
$$f'(0)=\exp(2\pi i \alpha(f))$$
where $\alpha(f)\in\C$ and $-\frac{1}{2} < \Re(\alpha(f)) \le \frac{1}{2}$ our study is focussed on the maps in the following class
$$ \mathcal{F}_1 = \{ f\in \mathcal{F}\colon |\arg(\alpha(f))| < \frac{\pi}{4}\} $$

=== Comments ===

By the definition we have $$
\begin{align*}
\log(f'(0))&=2\pi i\alpha(f)=2\pi i re^{i\beta}=2\pi r e^{i(\beta+\frac{\pi}{2})}\\
\arg(\log(f'(0))&=\beta+\frac{\pi}{2}
\end{align*}$$
so the condition for $\mathcal{F}_1$ is
$|\arg(\log(f'(0)))-\frac{\pi}{2}|<\frac{\pi}{4}$ or
$$\frac{\pi}{4} < \arg(\log(f'(0)) < \frac{3\pi}{4}$$

== $\sigma(f)$ ==
[p. 340]

Let $\sigma(f)$ be the other fixpoint of $f$ (besides 0).

...

$$\sigma(f) = 2\pi i \alpha(f)(1+o(1)), \quad\text{as}\quad f\to f_0$$

=== Comments ===

By the comment before, we can say that $\sigma(f)$ lies roughly in the sector of angle $(\frac{\pi}{4},\frac{3\pi}{4})$ above the tip 0.

== Proposition 3.2.2 ==
[p. 340]
Let $f_0\in\mathcal{F}_0$. There exists a neighborhood $\mathcal{N}_0$ of $f_0$ (in the sense of the topology defined in 2.5.1) such that if $f\in\mathcal{N}_0\cap \mathcal{F}_1$, then there exist closed Jordan domains $S_{-,f}$, $S_{+,f}$ and analytic functions $\Phi_{-,f}$, $\Phi_{+,f}$ satisfying the following:

(i) $S_{+,f}$ is bounded by an arc $\ell_{\pm}$ and its image $f(\ell_{\pm})$ such that $\ell_{\pm}$ joins two fixed points $\{0,\sigma(f)\}$, $\ell_{\pm}\cap f(\ell_{\pm}) = \{0,\sigma(f)\}$ and $S_{-,f}\cap S_{+,f} = \{0,\sigma(f)\}$;

(ii) $\Phi_{\pm,f}$ is defined, analytic and injective in a neighborhood of $S'_{\pm,f} = S_{\pm,f}\setminus \{0,\sigma(f)\}$;

(iii) if $z\in S'_{-,f}$ then there is an $n\ge 1$ such that $f^n(z)\in S'_{+,f}$ and for the smallest such $n$,
\[ \Phi_{+,f}(f^n(z)) = \Phi_{-,f}(z) - \frac{1}{\alpha(f)} + n \]

(iv) when $f\in\mathcal{F}_1$ tends to $f_0$, $S_{\pm,f}\to S_{\pm,0}\cup \{0\}$ in the Hausdorff metric, and $\Phi_{\pm,f}\to \Phi_{\pm,0}$ in the topology defined in 2.5.1.

== Proposition 4.4.1 ==
[p. 356]

Let $f_0\in\mathcal{F}_0$, there exists a neighborhood $\mathcal{N}_0$ of $f_0$ (in the previously described "compact-open topology with domain of definition"), such that if $f\in \mathcal{N}_0 \cap \mathcal{F}_1$, then there exist large $\xi_0, \eta_0>0$, and analytic maps $\ph_f\colon D(\ph_f)\to\overline{\C}$ and $\tilde{\mathcal{R}}_f\colon D(\tilde{\mathcal{R}}_f)\to \C$ satisfying the following:
# $D(\ph_f)$ contains the set $$ \mathcal{Q}_f = \{ w\in\C\colon |\arg(-w-\xi_0)| < \frac{2\pi}{3}, |\arg(w+\frac{1}{\alpha(f)}-\xi_0)|<\frac{2\pi}{3} \} $$ and $\ph_f(D(\ph_f)) \subseteq D(f)$; If $w, w+1\in D(\ph_f)$ then $$ \ph_f(w+1)=f(\ph_f(w)) $$ $\ph_f(w)\to 0$ when $w\in \mathcal{Q}_f$ and $\Im(w)\to\infty$; $\ph_f(w)\to\sigma(f)$ when $w\in\mathcal{Q}_f$ and $\Im(w)\to -\infty$.
# $D(\tilde{\mathcal{R}}_f)$ contains $\{ w\in\C\colon |\Im(w)| > \eta_0\}$ and is invariant under $T(w)=w+1$; if $w\in D(\tilde{\mathcal{R}}_f)$ then $$\tilde{\mathcal{R}}_f(w+1)=\tilde{\mathcal{R}}(w)+1;$$ $\tilde{\mathcal{R}}_f(w)-w$ tends to $-\frac{1}{\alpha(f)}$ when $\Im(w)\to\infty$; to a constant when $\Im(w)\to -\infty$.
# If $w\in D(\tilde{\mathcal{R}}_f)\cap D(\ph_f)$ and $w'=\tilde{\mathcal{R}}_f(w)+n\in D(\ph_f)$ for some $n\in\Z$ then $$f^n(\ph_f(w))=\ph_f(w')\quad \text{for}\quad n\ge 0, \quad f^{-n}(\ph_f(w'))=\ph_f(w)\quad \text{for}\quad n<0$$ Moreover if $|\arg(w'+\frac{1}{2\alpha(f)}-\xi_0)|<\frac{2\pi}{3}$ then $n>0$.
# When $f\in\mathcal{F_1}$ and $f\to f_0$ $$\ph_f\to\ph_0\quad\text{and}\quad \tilde{\mathcal{R}}_f + \frac{1}{\alpha(f)} \to \tilde{\mathcal{E}}_{f_0}$$

<references/>

Shishikura perturbed Fatou coordinates

2013-02-19T21:35:34Z

Bo198214: /* Proposition 3.2.2 */ fixed some TeX

Shishikura writes<ref name="Shishikura2000">Shishikura, M. (2000). Bifurcation of parabolic fixed points. In Lei, Tan, The Mandelbrot set, theme and variations. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 274, 325-363 (2000)</ref> the following

== $\mathcal{F}_0$ ==
[p. 327]

If $f_0''(0)\neq 0$ by another coordinate change we may assume that $f_0''(0)=1$, so define
$$ \mathcal{F}_0 = \{ f_0 \colon f_0 \quad\text{is defined and analytic in a neighborhood of 0}\quad f_0(0)=0, f_0'(0)=1, f_0''(0)=1\}$$

== Neighborhood of $f$ in the compact-open topology with domain of definition ==
[ p. 332 ]

In what follows a map $f$ is always associated with its domain of definition $D(f)$. That is two maps are considered as distinct if they have distinct domains, even if the one is an extension of the other. A neighborhood of an analytic map $f\colon D(f)\to\overline{\C}$ is a set containing
$$\{ g \colon g \quad\text{is analytic},\quad D(g)\supset K, \sup_{z\in K} d(f(z)-g(z))<\eps \} $$
where $K$ is a compact set in $D(f)$, $\eps>0$ and $d(.,.)$ is the spherical metric.
A sequence $(f_n)$ converges to $f$ in this topology if and only if for any compact set $K\subset D(f)$ there exists $n_0$ such that $K\subset D(f_n)$ for all $n\ge n_0$ and $f_n|_K\to f|_K$ uniformly on $K$ as $n\to\infty$. The system of these neighborhoods define the "compact-open topology together with the domain of defintion", which is unfortunately not Hausdorff, since an extension of $f$ is contained in any neighborhood of $f$.

For $b_1,b_2\in\C$ with $\Re(b_1)<\Re(b_2)$ define
$$Q(b_1,b_2)=\{z\in\C\colon \Re(z-b_1)>-|\Im(z-b_1)|, \Re(z-b_2)>|\Im(z-b_2)|\}$$
If $b_1=-\infty$ (resp. $b_2=\infty$) then the corresponding condition should be removed.

== Proposition 2.5.2 ==
[p. 333]

Let $f$ be a holomorphic function defined on $Q=Q(b_1,b_2)$ with $\Re(b_2)>\Re(b_1)+2$ (here $b_1$ or $b_2$ may be $-\infty$ or $\infty$). Suppose
$$|F(z)-(z+1)|<\frac{1}{4},\quad |F'(z)-(z+1)|<\frac{1}{4}\quad \text{for all}\quad z\in Q.$$

Then
# $F$ is univalent [means injective] on $Q$.
# Let $z_0\in Q$ be a point such that $\Re(b_1)<\Re(z_0)<\Re(b_2)-\frac{5}{4}$. Denote by $S$ the closed region (a strip) bounded by the two curves $\ell=\{z_0+i y\colon y\in\R\}$ and $F(\ell)$. Then for any $z\in Q$ there exist a unique $n\in \Z$ such that $F^n(z)$ is defined and belongs to $S-F(\ell)$.
# There exists an univalent function $\Phi\colon Q\to \C$ satisfying $$\Phi(F(z))=\Phi(z)+1$$ whenever both sides are defined. Moreover $\Phi$ is unique up to an addition of a constant.
# Fix a point $z_0\in Q$. If we normalize $\Phi$ by $\Phi(z_0)=0$ then the correspondence $F\mapsto \Phi$ is continuous with respect to the compact-open topology.

== $\mathcal{F}$, $\mathcal{F}_1$ and $\alpha(f)$ ==
[p. 339]

$$ \mathcal{F} = \{ f \colon f \quad\text{is defined and analytic in a neighborhood of 0}\quad f(0)=0, f_0'(0)\neq 0\}$$
For $f\in \mathcal{F}$ we express the derivative
$$f'(0)=\exp(2\pi i \alpha(f))$$
where $\alpha(f)\in\C$ and $-\frac{1}{2} < \Re(\alpha(f)) \le \frac{1}{2}$ our study is focussed on the maps in the following class
$$ \mathcal{F}_1 = \{ f\in \mathcal{F}\colon |\arg(\alpha(f))| < \frac{\pi}{4}\} $$

=== Comments ===

By the definition we have $$
\begin{align*}
\log(f'(0))&=2\pi i\alpha(f)=2\pi i re^{i\beta}=2\pi r e^{i(\beta+\frac{\pi}{2})}\\
\arg(\log(f'(0))&=\beta+\frac{\pi}{2}
\end{align*}$$
so the condition for $\mathcal{F}_1$ is
$|\arg(\log(f'(0)))-\frac{\pi}{2}|<\frac{\pi}{4}$ or
$$\frac{\pi}{4} < \arg(\log(f'(0)) < \frac{3\pi}{4}$$

== $\sigma(f)$ ==
[p. 340]

Let $\sigma(f)$ be the other fixpoint of $f$ (besides 0).

...

$$\sigma(f) = 2\pi i \alpha(f)(1+o(1)), \quad\text{as}\quad f\to f_0$$

=== Comments ===

By the comment before, we can say that $\sigma(f)$ lies roughly in the sector of angle $(\frac{\pi}{4},\frac{3\pi}{4})$ above the tip 0.

== Proposition 3.2.2 ==
Let $f_0\in\mathcal{F}_0$. There exists a neighborhood $\mathcal{N}_0$ of $f_0$ (in the sense of the topology defined in 2.5.1) such that if $f\in\mathcal{N}_0\cap \mathcal{F}_1$, then there exist closed Jordan domains $S_{-,f}$, $S_{+,f}$ and analytic functions $\Phi_{-,f}$, $\Phi_{+,f}$ satisfying the following:

(i) $S_{+,f}$ is bounded by an arc $\ell_{\pm}$ and its image $f(\ell_{\pm})$ such that $\ell_{\pm}$ joins two fixed points $\{0,\sigma(f)\}$, $\ell_{\pm}\cap f(\ell_{\pm}) = \{0,\sigma(f)\}$ and $S_{-,f}\cap S_{+,f} = \{0,\sigma(f)\}$;

(ii) $\Phi_{\pm,f}$ is defined, analytic and injective in a neighborhood of $S'_{\pm,f} = S_{\pm,f}\setminus \{0,\sigma(f)\}$;

(iii) if $z\in S'_{-,f}$ then there is an $n\ge 1$ such that $f^n(z)\in S'_{+,f}$ and for the smallest such $n$,
\[ \Phi_{+,f}(f^n(z)) = \Phi_{-,f}(z) - \frac{1}{\alpha(f)} + n \]

(iv) when $f\in\mathcal{F}_1$ tends to $f_0$, $S_{\pm,f}\to S_{\pm,0}\cup \{0\}$ in the Hausdorff metric, and $\Phi_{\pm,f}\to \Phi_{\pm,0}$ in the topology defined in 2.5.1.

== Proposition 4.4.1 ==
[p. 356]

Let $f_0\in\mathcal{F}_0$, there exists a neighborhood $\mathcal{N}_0$ of $f_0$ (in the previously described "compact-open topology with domain of definition"), such that if $f\in \mathcal{N}_0 \cap \mathcal{F}_1$, then there exist large $\xi_0, \eta_0>0$, and analytic maps $\ph_f\colon D(\ph_f)\to\overline{\C}$ and $\tilde{\mathcal{R}}_f\colon D(\tilde{\mathcal{R}}_f)\to \C$ satisfying the following:
# $D(\ph_f)$ contains the set $$ \mathcal{Q}_f = \{ w\in\C\colon |\arg(-w-\xi_0)| < \frac{2\pi}{3}, |\arg(w+\frac{1}{\alpha(f)}-\xi_0)|<\frac{2\pi}{3} \} $$ and $\ph_f(D(\ph_f)) \subseteq D(f)$; If $w, w+1\in D(\ph_f)$ then $$ \ph_f(w+1)=f(\ph_f(w)) $$ $\ph_f(w)\to 0$ when $w\in \mathcal{Q}_f$ and $\Im(w)\to\infty$; $\ph_f(w)\to\sigma(f)$ when $w\in\mathcal{Q}_f$ and $\Im(w)\to -\infty$.
# $D(\tilde{\mathcal{R}}_f)$ contains $\{ w\in\C\colon |\Im(w)| > \eta_0\}$ and is invariant under $T(w)=w+1$; if $w\in D(\tilde{\mathcal{R}}_f)$ then $$\tilde{\mathcal{R}}_f(w+1)=\tilde{\mathcal{R}}(w)+1;$$ $\tilde{\mathcal{R}}_f(w)-w$ tends to $-\frac{1}{\alpha(f)}$ when $\Im(w)\to\infty$; to a constant when $\Im(w)\to -\infty$.
# If $w\in D(\tilde{\mathcal{R}}_f)\cap D(\ph_f)$ and $w'=\tilde{\mathcal{R}}_f(w)+n\in D(\ph_f)$ for some $n\in\Z$ then $$f^n(\ph_f(w))=\ph_f(w')\quad \text{for}\quad n\ge 0, \quad f^{-n}(\ph_f(w'))=\ph_f(w)\quad \text{for}\quad n<0$$ Moreover if $|\arg(w'+\frac{1}{2\alpha(f)}-\xi_0)|<\frac{2\pi}{3}$ then $n>0$.
# When $f\in\mathcal{F_1}$ and $f\to f_0$ $$\ph_f\to\ph_0\quad\text{and}\quad \tilde{\mathcal{R}}_f + \frac{1}{\alpha(f)} \to \tilde{\mathcal{E}}_{f_0}$$

<references/>

Shishikura perturbed Fatou coordinates

2013-02-19T21:32:18Z

Bo198214: added new section from Shishikura

Shishikura writes<ref name="Shishikura2000">Shishikura, M. (2000). Bifurcation of parabolic fixed points. In Lei, Tan, The Mandelbrot set, theme and variations. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 274, 325-363 (2000)</ref> the following

== $\mathcal{F}_0$ ==
[p. 327]

If $f_0''(0)\neq 0$ by another coordinate change we may assume that $f_0''(0)=1$, so define
$$ \mathcal{F}_0 = \{ f_0 \colon f_0 \quad\text{is defined and analytic in a neighborhood of 0}\quad f_0(0)=0, f_0'(0)=1, f_0''(0)=1\}$$

== Neighborhood of $f$ in the compact-open topology with domain of definition ==
[ p. 332 ]

In what follows a map $f$ is always associated with its domain of definition $D(f)$. That is two maps are considered as distinct if they have distinct domains, even if the one is an extension of the other. A neighborhood of an analytic map $f\colon D(f)\to\overline{\C}$ is a set containing
$$\{ g \colon g \quad\text{is analytic},\quad D(g)\supset K, \sup_{z\in K} d(f(z)-g(z))<\eps \} $$
where $K$ is a compact set in $D(f)$, $\eps>0$ and $d(.,.)$ is the spherical metric.
A sequence $(f_n)$ converges to $f$ in this topology if and only if for any compact set $K\subset D(f)$ there exists $n_0$ such that $K\subset D(f_n)$ for all $n\ge n_0$ and $f_n|_K\to f|_K$ uniformly on $K$ as $n\to\infty$. The system of these neighborhoods define the "compact-open topology together with the domain of defintion", which is unfortunately not Hausdorff, since an extension of $f$ is contained in any neighborhood of $f$.

For $b_1,b_2\in\C$ with $\Re(b_1)<\Re(b_2)$ define
$$Q(b_1,b_2)=\{z\in\C\colon \Re(z-b_1)>-|\Im(z-b_1)|, \Re(z-b_2)>|\Im(z-b_2)|\}$$
If $b_1=-\infty$ (resp. $b_2=\infty$) then the corresponding condition should be removed.

== Proposition 2.5.2 ==
[p. 333]

Let $f$ be a holomorphic function defined on $Q=Q(b_1,b_2)$ with $\Re(b_2)>\Re(b_1)+2$ (here $b_1$ or $b_2$ may be $-\infty$ or $\infty$). Suppose
$$|F(z)-(z+1)|<\frac{1}{4},\quad |F'(z)-(z+1)|<\frac{1}{4}\quad \text{for all}\quad z\in Q.$$

Then
# $F$ is univalent [means injective] on $Q$.
# Let $z_0\in Q$ be a point such that $\Re(b_1)<\Re(z_0)<\Re(b_2)-\frac{5}{4}$. Denote by $S$ the closed region (a strip) bounded by the two curves $\ell=\{z_0+i y\colon y\in\R\}$ and $F(\ell)$. Then for any $z\in Q$ there exist a unique $n\in \Z$ such that $F^n(z)$ is defined and belongs to $S-F(\ell)$.
# There exists an univalent function $\Phi\colon Q\to \C$ satisfying $$\Phi(F(z))=\Phi(z)+1$$ whenever both sides are defined. Moreover $\Phi$ is unique up to an addition of a constant.
# Fix a point $z_0\in Q$. If we normalize $\Phi$ by $\Phi(z_0)=0$ then the correspondence $F\mapsto \Phi$ is continuous with respect to the compact-open topology.

== $\mathcal{F}$, $\mathcal{F}_1$ and $\alpha(f)$ ==
[p. 339]

$$ \mathcal{F} = \{ f \colon f \quad\text{is defined and analytic in a neighborhood of 0}\quad f(0)=0, f_0'(0)\neq 0\}$$
For $f\in \mathcal{F}$ we express the derivative
$$f'(0)=\exp(2\pi i \alpha(f))$$
where $\alpha(f)\in\C$ and $-\frac{1}{2} < \Re(\alpha(f)) \le \frac{1}{2}$ our study is focussed on the maps in the following class
$$ \mathcal{F}_1 = \{ f\in \mathcal{F}\colon |\arg(\alpha(f))| < \frac{\pi}{4}\} $$

=== Comments ===

By the definition we have $$
\begin{align*}
\log(f'(0))&=2\pi i\alpha(f)=2\pi i re^{i\beta}=2\pi r e^{i(\beta+\frac{\pi}{2})}\\
\arg(\log(f'(0))&=\beta+\frac{\pi}{2}
\end{align*}$$
so the condition for $\mathcal{F}_1$ is
$|\arg(\log(f'(0)))-\frac{\pi}{2}|<\frac{\pi}{4}$ or
$$\frac{\pi}{4} < \arg(\log(f'(0)) < \frac{3\pi}{4}$$

== $\sigma(f)$ ==
[p. 340]

Let $\sigma(f)$ be the other fixpoint of $f$ (besides 0).

...

$$\sigma(f) = 2\pi i \alpha(f)(1+o(1)), \quad\text{as}\quad f\to f_0$$

=== Comments ===

By the comment before, we can say that $\sigma(f)$ lies roughly in the sector of angle $(\frac{\pi}{4},\frac{3\pi}{4})$ above the tip 0.

== Proposition 3.2.2 ==
Let $f_0\in\mathcal{F}_0$. There exists a neighborhood $\mathcal{N}$ of $f_0$ (in the sense of the topology defined in 2.5.1) such that if $f\in\mathcal{N}_0\cap \mathcal{F}_1, then there exist closed Jordan domains $S_{-,f}$, $S_{+,f}$ and analytic functions $\Phi_{-,f}$, $\Phi_{+,f}$ satisfying the following:

(i) $S_{+,f}$ is bounded by an arc $\ell_{\pm}$ and its image $f(\ell_{\pm})$ such that $\ell_{\pm}$ joins two fixed points $\{0,\sigma(f)\}$, $\ell_{\pm}\cap f(\ell_{\pm}) = \{0,\sigma(f)\}$ and $S_{-,f}\cap S_{+,f} = \{0,\sigma(f)\}$;

(ii) $\Phi_{\pm,f}$ is defined, analytic and injective in a neighborhood of $S'_{\pm,f} = S_{\pm,f}\setminus \{0,\sigma(f)\}$;

(iii) if $z\in S'_{-,f}$ then there is an $n\ge 1$ such that $f^n(z)\in S'_{+,f}$ and for the smallest such $n$,
\[ \Phi_{+,f}(f^n(z)) = \Phi_{-,f}(z) - \frac{1}{\alpha(f)} + n \]

(iv) when $f\in\mathcal{F}_1$ tends to $f_0$, $S_{\pm,f}\to S_{\pm,0}\cup \{0\}$ in the Hausdorff metric, and $\Phi_{\pm,f}\to \Phi_{\pm,0}$ in the topology defined in 2.5.1.

== Proposition 4.4.1 ==
[p. 356]

Let $f_0\in\mathcal{F}_0$, there exists a neighborhood $\mathcal{N}_0$ of $f_0$ (in the previously described "compact-open topology with domain of definition"), such that if $f\in \mathcal{N}_0 \cap \mathcal{F}_1$, then there exist large $\xi_0, \eta_0>0$, and analytic maps $\ph_f\colon D(\ph_f)\to\overline{\C}$ and $\tilde{\mathcal{R}}_f\colon D(\tilde{\mathcal{R}}_f)\to \C$ satisfying the following:
# $D(\ph_f)$ contains the set $$ \mathcal{Q}_f = \{ w\in\C\colon |\arg(-w-\xi_0)| < \frac{2\pi}{3}, |\arg(w+\frac{1}{\alpha(f)}-\xi_0)|<\frac{2\pi}{3} \} $$ and $\ph_f(D(\ph_f)) \subseteq D(f)$; If $w, w+1\in D(\ph_f)$ then $$ \ph_f(w+1)=f(\ph_f(w)) $$ $\ph_f(w)\to 0$ when $w\in \mathcal{Q}_f$ and $\Im(w)\to\infty$; $\ph_f(w)\to\sigma(f)$ when $w\in\mathcal{Q}_f$ and $\Im(w)\to -\infty$.
# $D(\tilde{\mathcal{R}}_f)$ contains $\{ w\in\C\colon |\Im(w)| > \eta_0\}$ and is invariant under $T(w)=w+1$; if $w\in D(\tilde{\mathcal{R}}_f)$ then $$\tilde{\mathcal{R}}_f(w+1)=\tilde{\mathcal{R}}(w)+1;$$ $\tilde{\mathcal{R}}_f(w)-w$ tends to $-\frac{1}{\alpha(f)}$ when $\Im(w)\to\infty$; to a constant when $\Im(w)\to -\infty$.
# If $w\in D(\tilde{\mathcal{R}}_f)\cap D(\ph_f)$ and $w'=\tilde{\mathcal{R}}_f(w)+n\in D(\ph_f)$ for some $n\in\Z$ then $$f^n(\ph_f(w))=\ph_f(w')\quad \text{for}\quad n\ge 0, \quad f^{-n}(\ph_f(w'))=\ph_f(w)\quad \text{for}\quad n<0$$ Moreover if $|\arg(w'+\frac{1}{2\alpha(f)}-\xi_0)|<\frac{2\pi}{3}$ then $n>0$.
# When $f\in\mathcal{F_1}$ and $f\to f_0$ $$\ph_f\to\ph_0\quad\text{and}\quad \tilde{\mathcal{R}}_f + \frac{1}{\alpha(f)} \to \tilde{\mathcal{E}}_{f_0}$$

<references/>

Hyperbolic fixpoint

2011-06-09T18:11:11Z

Bo198214: forgot some terms

A hyperbolic fixpoint is a [[fixpoint]] $a$ of $f$ such that $|f'(a)|\neq 0,1$.

For a locally analytic function with hyperbolic fixpoint at $0$, i.e. $f(z)=c_1 z + c_2z^2 + \dots$, $|c_1|\neq 0,1$, there is always a locally analytic and injective function $\sigma$ that satisfies the [[Schröder equation]]
$$\sigma(f(z))=c_1 \sigma(z))$$
$\sigma$ is unique up to a multiplicative constant and is called the [[Schröder coordinates]] of $f$ at 0.

Schröder iterate

2011-06-08T08:39:23Z

Bo198214: changed to coordinate

A way to compute regular iterates of a function $f$ with [[attracting fixpoint|attracting]] or [[hyperbolic fixpoint]] $a$.

For $z$ in the immediate basin of attraction of the attracting fixpoint $a$ and $f'(a)=c_1$, it is:
$$f^t(z)=\lim_{n\to\infty} f^{-n}(c_1^t f^n(z))$$

Or $\sigma$ being the [[Schröder coordinate]] of $f$ at $a$, we have:
$$f^t(z)=\sigma^{-1}(c_1^t \sigma(z))$$

Böttcher iterate

2011-06-08T08:38:22Z

Bo198214: first draft

A way to compute the regular iterates of a function $f$ with [[super-attracting fixpoint]] $a$.

For $f(z)=c_m (z-a)^m + c_{m+1} (z-a)^{m+1} + \dots$ the Böttcher iterate is:
$$f^t(z)=\lim_{n\to\infty} f^{-n}(f^n(z)^{m^t})$$

Or if $\beta$ is the [[Böttcher coordinate]] of $f$ at $a$, then
$$f^t(z)=\beta^{-1}\left(\beta(z)^{m^t}\right)$$